Convergence (mathematics)From Wikipedia, the free encyclopediaContents1 Absolute convergence 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Relation to convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Proof that any absolutely convergent series of complex numbers is convergent . . . . . . . . 21.2.2 Proof that any absolutely convergent series in a Banach space is convergent . . . . . . . . . 21.3 Rearrangements and unconditional convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Proof of the theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Products of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Absolute convergence of integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Addition chain 62.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Methods for computing addition chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Chain length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Brauer chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Scholz conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Addition-chain exponentiation 93.1 Addition-subtractionchain exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Addition-subtraction chain 114.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Almost convergent sequence 135.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Arithmetic progression 14iii CONTENTS6.1 Sum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.1.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.3 Standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.5 Formulas at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Betti number 187.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2 Example 1: Betti numbers of a simplicial complex K. . . . . . . . . . . . . . . . . . . . . . . . . 197.3 Example 2: the rst Betti number in graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.6 Relationship with dimensions of spaces of dierential forms . . . . . . . . . . . . . . . . . . . . . 217.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Cauchy product 238.1 Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.1.1 Cauchy product of two nite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.1.2 Cauchy product of two innite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.1.3 Cauchy product of two nite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.1.4 Cauchy product of two innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.1.5 Cauchy product of two power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3 Convergence and Mertens theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3.2 Proof of Mertens theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.4.1 Finite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.4.2 Innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.5 Cesros theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.5.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.6.1 Products of nitely many innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.7 Relation to convolution of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28CONTENTS iii9 Cauchy sequence 309.1 In real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.3.2 Counter-example: rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.3.3 Counter-example: open interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.3.4 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.4.1 In topological vector spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.4.2 In topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.4.3 In groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.4.4 In constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.4.5 In a hyperreal continuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410Chebyshevs sum inequality 3510.1Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.2Continuous version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611Compact convergence 3711.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3812Complementary sequences 3912.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.3Properties of complementary pairs of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.4Golay pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.5Applications of complementary sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4213Conditional convergence 4313.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4313.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43iv CONTENTS13.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314Convergence in measure 4414.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.3Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4514.4Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4514.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4615Convergence of random variables 4715.1Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4715.2Convergence in distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4815.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4815.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4915.3Convergence in probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4915.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015.4Almost sure convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015.4.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5115.5Sure convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5115.6Convergence in mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5215.7Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5215.8See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.9Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516Convergence problem 5616.1Elementary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.1.1 Periodic continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.1.2 The special case when period k = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.1.3 Worpitzkys theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.2leszyskiPringsheim criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.3Van Vlecks theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6017Convergent series 6117.1Examples of convergent and divergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.2Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.3Conditional and absolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6317.4Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6417.5Cauchy convergence criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65CONTENTS v17.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6617.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6618Cumulant-generating function 6718.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.1.1 Alternative denition of the cumulant generating function . . . . . . . . . . . . . . . . . . 6718.2Uses in statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.3Cumulants of some discrete probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . 6818.4Cumulants of some continuous probability distributions . . . . . . . . . . . . . . . . . . . . . . . 6918.5Some properties of the cumulant generating function . . . . . . . . . . . . . . . . . . . . . . . . . 6918.6Some properties of cumulants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.6.1 Invariance and equivariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.6.2 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.6.3 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.6.4 A negative result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.6.5 Cumulants and moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7118.6.6 Relation to moment-generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . 7318.6.7 Cumulants and set-partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7318.6.8 Cumulants and combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7418.7Joint cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7418.7.1 Conditional cumulants and the law of total cumulance . . . . . . . . . . . . . . . . . . . . 7518.8Relation to statistical physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.9History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7618.10Cumulants in generalized settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7618.10.1 Formal cumulants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7618.10.2 Bell numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.10.3 Cumulants of a polynomial sequence of binomial type . . . . . . . . . . . . . . . . . . . . 7718.10.4 Free cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7818.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7819Cutting sequence 7919.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7920Cyclic sieving 8020.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8020.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8020.3Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8121DavenportSchinzel sequence 8221.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221.2Length bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82vi CONTENTS21.3Application to lower envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8321.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8321.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8421.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8421.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8522Disjunctive sequence 8622.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8622.2Rich numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8722.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8722.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8823Divisibility sequence 8923.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8923.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8924Ducci sequence 9124.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9124.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9124.3Modulo two form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9224.4Cellular automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9224.5Other related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9224.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9324.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9325Examples of generating functions 9425.1Worked example A: basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9425.1.1 Bivariate generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9425.2Worked example B: Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9425.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9526Factorial moment generating function 9626.1Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9626.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9727Farey sequence 9827.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9827.2History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9827.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9927.3.1 Sequence length and index of a fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9927.3.2 Farey neighbours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10227.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10327.3.4 Ford circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10327.3.5 Riemann hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103CONTENTS vii27.4Next term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10327.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10527.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10527.7Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10527.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10628Flat convergence 10728.1Integral currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10728.2Flat norm and at distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10728.3Compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10728.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10729Generating function 10929.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10929.1.1 Ordinary generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10929.1.2 Exponential generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11029.1.3 Poisson generating function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11029.1.4 Lambert series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11029.1.5 Bell series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11029.1.6 Dirichlet series generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11129.1.7 Polynomial sequence generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . 11129.2Ordinary generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11129.2.1 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11229.2.2 Multiplication yields convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11329.2.3 Relation to discrete-time Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . 11329.2.4 Asymptotic growth of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11329.2.5 Bivariate and multivariate generating functions . . . . . . . . . . . . . . . . . . . . . . . 11429.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.3.1 Ordinary generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.3.2 Exponential generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.3.3 Bell series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.3.4 Dirichlet series generating function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.3.5 Multivariate generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.4.1 Techniques of evaluating sums with generating function . . . . . . . . . . . . . . . . . . . 11629.4.2 Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11629.4.3 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.5Other generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11829.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11829.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11829.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119viii CONTENTS30Geometric progression 12030.1Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12130.2Geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12230.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12230.2.2 Related formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12230.2.3 Innite geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12330.2.4 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12530.3Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12530.4Relationship to geometry and Euclids work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12630.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12630.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12730.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12731GromovHausdor convergence 12831.1GromovHausdor distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12831.2Some properties of GromovHausdor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12831.3Pointed GromovHausdor convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12831.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12831.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12932Halton sequence 13032.1Example of Halton sequence used to generate points in (0, 1) (0, 1) in R2. . . . . . . . . . . . . 13032.2Implementation in Pseudo Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13232.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13232.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13232.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13233Harmonic progression (mathematics) 13333.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13333.2Use in geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13333.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13333.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13434Innite product 13534.1Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13534.2Product representations of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13634.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13634.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13734.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13735Interleave sequence 13835.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138CONTENTS ix36Intrinsic at distance 13936.1Intrinsic at distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13936.2Riemannian setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13936.3Integral current spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14036.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14036.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14036.6Citations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037Iterated function 14237.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14237.2Abelian property and Iteration sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14237.3Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.4Limiting behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.5Fractional iterates and ows, and negative iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.6Some formulas for fractional iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14437.6.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14437.6.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14437.6.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14537.7Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14537.8Markov chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14537.9Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14637.10Means of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14637.11In computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14637.12Denitions in terms of iterated functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14637.13Lies data transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14637.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14737.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14738Katydid sequence 14838.1Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14838.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14838.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14839Limit of a sequence 14939.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14939.2Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15039.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15039.2.2 Formal Denition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15139.2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15139.2.4 Innite limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15139.3Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152x CONTENTS39.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.4Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.4.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.5Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.6Denition in hyperreal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15239.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15339.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15339.8.1 Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15339.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15439.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15440List of sums of reciprocals 15540.1Finitely many terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15540.2Innitely many terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15640.2.1 Convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15640.2.2 Divergent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15740.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15740.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15741Logarithmically concave sequence 15841.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15841.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15842Low-discrepancy sequence 15942.1Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15942.1.1 Low-discrepancy sequences in numerical integration . . . . . . . . . . . . . . . . . . . . 15942.2Denition of discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16042.3The KoksmaHlawka inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16142.4The formula of Hlawka-Zaremba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16142.5The L2version of the KoksmaHlawka inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 16142.6The ErdsTurnKoksma inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16242.7The main conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16242.8Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16342.9Construction of low-discrepancy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16342.9.1 Random numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16342.9.2 Additive recurrence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16442.9.3 Sobol sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16442.9.4 van der Corput sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16542.9.5 Halton sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16542.9.6 Hammersley set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16542.9.7 Poisson disk sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166CONTENTS xi42.10Graphical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16642.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16642.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16643Mathematics of oscillation 17443.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17443.1.1 Oscillation of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17443.1.2 Oscillation of a function on an open set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17543.1.3 Oscillation of a function at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17543.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17543.3Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17543.4Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17743.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17743.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17744Matsushimas formula 17844.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17845Modes of convergence 17945.1Elements of a topological space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17945.2Series of elements in a topological abelian group . . . . . . . . . . . . . . . . . . . . . . . . . . . 17945.3Convergence of sequence of functions on a topological space . . . . . . . . . . . . . . . . . . . . . 17945.4Series of functions on a topological abelian group . . . . . . . . . . . . . . . . . . . . . . . . . . . 18045.5Functions dened on a measure space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18045.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18046Modes of convergence (annotated index) 18146.1A sequence of elements {an} in a topological space (Y) . . . . . . . . . . . . . . . . . . . . . . . 18146.1.1 ...in a uniform space (U) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18146.2A series of elements bk in a TAG (G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18146.2.1 ...in a normed space (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18246.3A sequence of functions {fn} from a set (S) to a topological space (Y) . . . . . . . . . . . . . . . . 18246.3.1 ...from a set (S) to a uniform space (U). . . . . . . . . . . . . . . . . . . . . . . . . . . . 18246.4A series of functions gk from a set (S) to a TAG (G) . . . . . . . . . . . . . . . . . . . . . . . . 18346.4.1 ...from a set (S) to a normed space (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18346.4.2 ...from a topological space (X) to a TAG (G) . . . . . . . . . . . . . . . . . . . . . . . . . 18346.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18447Moment-generating function 18547.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18547.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18647.3Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18647.3.1 Sum of independent random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186xii CONTENTS47.3.2 Vector-valued random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18647.4Important properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18747.4.1 Calculations of moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18747.5Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18747.6Relation to other functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18747.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18847.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18848Monotone convergence theorem 18948.1Convergence of a monotone sequence of real numbers . . . . . . . . . . . . . . . . . . . . . . . . 18948.1.1 Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18948.1.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18948.1.3 Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18948.1.4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18948.1.5 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18948.1.6 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18948.2Convergence of a monotone series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19048.2.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19048.3Lebesgues monotone convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19048.3.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19048.3.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19148.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19348.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19349Normal convergence 19449.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19449.2Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19449.3Distinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19449.4Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19549.4.1 Local normal convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19549.4.2 Compact normal convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19549.5Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19549.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19549.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19550Periodic sequence 19650.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19650.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19650.3Periodic 0, 1 sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19750.4Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19751Pointwise convergence 19851.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198CONTENTS xiii51.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19851.3Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19951.4Almost everywhere convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19951.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19951.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19952Polynomial sequence 20052.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20052.2Classes of polynomial sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20152.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20152.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20153Polyphase sequence 20253.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20253.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20254Probability-generating function 20354.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20354.1.1 Univariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20354.1.2 Multivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20354.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20354.2.1 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20354.2.2 Probabilities and expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20454.2.3 Functions of independent random variables . . . . . . . . . . . . . . . . . . . . . . . . . . 20454.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20554.4Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20654.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20654.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20655Radius of convergence 20755.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20755.2Finding the radius of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20755.2.1 Theoretical radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20855.2.2 Practical estimation of radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20855.3Radius of convergence in complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20955.3.1 A simple example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21055.3.2 A more complicated example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21055.4Convergence on the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21155.5Comments on rate of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21255.6A graphical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21255.7Abscissa of convergence of a Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21255.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21255.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213xiv CONTENTS55.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21356Random sequence 21456.1Early history. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21556.2Modern approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21556.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21556.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21656.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21656.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21657Rook polynomial 21757.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21757.1.1 Complete boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21857.2Matching polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21857.3Connection to matrix permanents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21857.4Complete rectangular boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21857.4.1 Rooks problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21857.4.2 The rook polynomial as a generalization of the rooks problem. . . . . . . . . . . . . . . . 21957.4.3 Symmetric arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22057.4.4 Arrangements counted by symmetry classes . . . . . . . . . . . . . . . . . . . . . . . . . 22157.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22158Scarborough criterion 22358.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22358.2Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22358.3GaussSeidel method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22358.4Diagonal dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22358.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22458.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22458.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22459Scholz conjecture 22559.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22559.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22560Sequence 22660.1Examples and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22760.1.1 Important examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22760.1.2 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22860.1.3 Specifying a sequence by recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22960.2Formal denition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22960.2.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22960.2.2 Finite and innite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230CONTENTS xv60.2.3 Increasing and decreasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23060.2.4 Bounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23060.2.5 Other types of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23060.3Limits and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23160.3.1 Denition of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23260.3.2 Applications and important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23260.3.3 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23360.4Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23360.5Use in other elds of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23460.5.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23460.5.2 Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23460.5.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23560.5.4 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23560.5.5 Set theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23660.5.6 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23660.5.7 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23660.6Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23660.7Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23760.8Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23760.9See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23760.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23760.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23861Sequence space 23961.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23961.1.1 pspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23961.1.2 c and c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24061.1.3 Other sequence spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24061.2Properties of pspaces and the space c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24161.2.1 pspaces are increasing in p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24261.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24261.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24262Shift rule 24362.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24363Sobol sequence 24463.1Good distributions in the s-dimensional unit hypercube . . . . . . . . . . . . . . . . . . . . . . . 24463.2A fast algorithm for the construction of Sobol sequences . . . . . . . . . . . . . . . . . . . . . . . 24563.3Additional uniformity properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24563.4The initialization of Sobol numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24663.5Implementation and availability of Sobol sequences . . . . . . . . . . . . . . . . . . . . . . . . . 246xvi CONTENTS63.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24663.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24763.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24763.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24764Stationary sequence 24864.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24864.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24865Sturmian word 24965.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24965.1.1 Combinatoric denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24965.1.2 Geometric denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25065.2Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25065.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25065.2.2 Balanced aperiodic sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25165.2.3 Slope and intercept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25165.2.4 Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25265.3Non-binary words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25265.4Associated real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25265.5History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25265.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25265.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25365.8Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25366Subadditivity 25466.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25466.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25466.3Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25566.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25566.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25566.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25566.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25667Subsequence 25767.1Common subsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25767.2Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25767.3Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25867.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25867.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25868Subsequential limit 25969Superadditivity 260CONTENTS xvii69.1Examples of superadditive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26069.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26069.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26170Tuple 26270.1Etymology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26270.1.1 Names for tuples of specic lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26270.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26270.3Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26370.3.1 Tuples as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26370.3.2 Tuples as nested ordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26370.3.3 Tuples as nested sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26470.4 n-tuples of m-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26470.5Type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26470.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26570.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26570.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26671Unconditional convergence 26771.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26771.2Alternative denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26771.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26771.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26772Uniform absolute-convergence 26972.1Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26972.2Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26972.3Distinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26972.4Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27072.5Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27072.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27072.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27073Uniform convergence 27173.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27173.2Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27173.2.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27273.2.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27273.2.3 Denition in a hyperreal setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27273.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27273.3.1 Exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27373.4Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27373.5Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274xviii CONTENTS73.5.1 To continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27473.5.2 To dierentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27573.5.3 To integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27573.5.4 To analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27673.5.5 To series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27673.6Almost uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27673.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27673.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27673.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27673.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27774Uniformly Cauchy sequence 27874.1Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27874.2Generalization to uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27874.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27875Van der Corput sequence 27975.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28075.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28075.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28075.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28076Vectorial addition chain 28176.1Addition sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28176.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28276.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28277Vites formula 28377.1Signicance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28477.2Interpretation and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28477.3Related formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28577.4Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28577.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28677.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28778Weisners method 28878.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28878.2Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 28978.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28978.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29578.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297Chapter 1Absolute convergenceIn mathematics, an innite series of numbers is said to converge absolutely (or to be absolutely convergent) if thesum of the absolute value of the summand is nite. More precisely, a real or complex series n=0an is said toconverge absolutely if n=0|an| =L for some real numberL . Similarly, an improper integral of a function,0f(x) dx , is said to converge absolutely if the integral of the absolute value of the integrand is nitethat is, if0|f(x)| dx = L.Absolute convergence is important for the study of innite series because its denition is strong enough to haveproperties of nite sums that not all convergent series possess, yet is broad enough to occur commonly. (A convergentseries that is not absolutely convergent is called conditionally convergent.)1.1 BackgroundOne may study the convergence of seriesn=0an whose terms an are elements of an arbitrary abelian topologicalgroup. The notion of absolute convergence requires more structure, namely a norm, which is a real-valued function : G R on abelian group G (written additively, with identity element 0) such that:1. The norm of the identity element of G is zero: 0 = 0.2. For every x in G, x = 0 implies x = 0.3. For every x in G, x = x.4. For every x, y in G, x +y x +y.In this case, the function d(x, y) = x y induces on G the structure of a metric space (a type of topology). Wecan therefore consider G-valued series and dene such a series to be absolutely convergent ifn=0an < .In particular, these statements apply using the norm |x| (absolute value) in the space of real numbers or complexnumbers.1.2 Relation to convergenceIf G is complete with respect to the metric d, then every absolutely convergent series is convergent. The proof is thesame as for complex-valued series: use the completeness to derive the Cauchy criterion for convergencea series isconvergent if and only if its tails can be made arbitrarily small in normand apply the triangle inequality.In particular, for series with values in any Banach space, absolute convergence implies convergence. The converse isalso true: if absolute convergence implies convergence in a normed space, then the space is a Banach space.If a series is convergent but not absolutely convergent, it is called conditionally convergent. An example of a con-ditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence,most notably including the ratio test and the root test, demonstrate absolute convergence. This is because a powerseries is absolutely convergent on the interior of its disk of convergence.12 CHAPTER 1. ABSOLUTE CONVERGENCE1.2.1 Proof that any absolutely convergent series of complex numbers is convergentSince a series of complex numbers converges if and only if both its real and imaginary parts converge, we may assumewith equal generality that the an are real numbers. Suppose that|an| is convergent. Then, 2|an| is convergent.Since 0 an +|an| 2|an| , we have0 mn=1(an +|an|) mn=12|an| n=12|an|Thus,mn=1(an +|an|) is a bounded monotonic sequence (in m), which must converge.an=(an +|an|) |an| is a dierence of convergent series; therefore, it is also convergent, as desired.1.2.2 Proof that any absolutely convergent series in a Banach space is convergentThe above result can be easily generalized to every Banach space (X, ). Let xn be an absolutely convergent seriesin X. Asnk=1 xk is a Cauchy sequence of real numbers, for any > 0 and large enough natural numbers m > n itholds:mk=1xk nk=1xk =mk=n+1xk < .By the triangle inequality for the norm , one immediately gets:_____mk=1xk nk=1xk_____ =_____mk=n+1xk_____ mk=n+1xk < ,which means thatnk=1 xk is a Cauchy sequence in X, hence the series is convergent in X.[1]1.3 Rearrangements and unconditional convergenceIn the general context of a G-valued series, a distinction is made between absolute and unconditional convergence, andthe assertion that a real or complex series which is not absolutely convergent is necessarily conditionally convergent(meaning not unconditionally convergent) is then a theorem, not a denition. This is discussed in more detail below.Given a series n=0an with values in a normed abelian group G and a permutation of the natural numbers, onebuilds a newseriesn=0a(n) , said to be a rearrangement of the original series. Aseries is said to be unconditionallyconvergent if all rearrangements of the series are convergent to the same value.When G is complete, absolute convergence implies unconditional convergence:Theorem. Leti=1ai= A G,i=1ai < and let : N N be a permutation. Then:i=1a(i)= A.The issue of the converse is interesting. For real series it follows from the Riemann rearrangement theorem thatunconditional convergence implies absolute convergence. Since a series with values in a nite-dimensional normed1.3. REARRANGEMENTS AND UNCONDITIONAL CONVERGENCE 3space is absolutely convergent if each of its one-dimensional projections is absolutely convergent, it follows thatabsolute and unconditional convergence coincide for Rn-valued series.But there are unconditionally and non-absolutely convergent series with values in Hilbert space 2, for example:an=1nen,where {en}n=1 is an orthonormal basis. A theorem of A. Dvoretzky and C. A. Rogers[2] asserts that every innite-dimensional Banach space admits an unconditionally convergent series that is not absolutely convergent.1.3.1 Proof of the theoremFor any > 0, we can choose some , N , such that:N> n=Nan _____Nn=1an A_____ M, letI,= {1, . . . , N} \ 1({1, . . . , N})S,= min {(k) : k I,}L,= max {(k) : k I,}Then_____Ni=1a(i) A_____ =______i1({1,...,N})a(i) A+iI,a(i)____________Nj=1aj A______+______iI,a(i)____________Nj=1aj A______+iI,__a(i)________Nj=1aj A______+L,j=S,aj______Nj=1aj A______+j=N+1aj S, N + 1< This shows that4 CHAPTER 1. ABSOLUTE CONVERGENCE > 0, M,, N> M,_____Ni=1a(i) A_____ < ,that is:i=1a(i)= AQ.E.D.1.4 Products of seriesThe Cauchy product of two series converges to the product of the sums if at least one of the series converges absolutely.That is, suppose thatn=0an= A andn=0bn= B .The Cauchy product is dened as the sum of terms cn where:cn=nk=0akbnk.Then, if either the an or bn sum converges absolutely, thenn=0cn= AB.1.5 Absolute convergence of integralsThe integralAf(x) dx of a real or complex-valued function is said to converge absolutely ifA|f(x)|dx < .One also says that f is absolutely integrable.When A = [a,b] is a closed bounded interval, every continuous function is integrable, and since f continuous implies|f| continuous, similarly every continuous function is absolutely integrable. It is not generally true that absolutelyintegrable functions on [a,b] are integrable: let S [a, b] be a nonmeasurable subset and take f= S1/2, whereS is the characteristic function of S. Then f is not Lebesgue measurable but |f| is constant. However, it is a standardresult that if f is Riemann integrable, so is |f|. This holds also for the Lebesgue integral; see below. On the otherhand a function f may be Kurzweil-Henstock integrable (or gauge integrable) while |f| is not. This includes thecase of improperly Riemann integrable functions.Similarly, when A is an interval of innite length it is well known that there are improperly Riemann integrablefunctions f which are not absolutely integrable. Indeed, given any series n=0an one can consider the associatedstep function fa:[0, ) R dened by fa([n, n + 1))=an . Then 0fadx converges absolutely, convergesconditionally or diverges according to the corresponding behavior ofn=0an.Another example of a convergent but not absolutely convergent improper Riemann integral isRsin xxdx .On any measure space A, the Lebesgue integral of a real-valued function is dened in terms of its positive and negativeparts, so the facts:1. f integrable implies |f| integrable1.6. SEE ALSO 52. f measurable, |f| integrable implies f integrableare essentially built into the denition of the Lebesgue integral. In particular, applying the theory to the countingmeasure on a set S, one recovers the notion of unordered summation of series developed by MooreSmith using(what are now called) nets. When S = N is the set of natural numbers, Lebesgue integrability, unordered summabilityand absolute convergence all coincide.Finally, all of the above holds for integrals with values in a Banach space. The denition of a Banach-valued Riemannintegral is an evident modication of the usual one. For the Lebesgue integral one needs to circumvent the decom-position into positive and negative parts with Daniells more functional analytic approach, obtaining the Bochnerintegral.1.6 See alsoConvergence of Fourier seriesConditional convergenceModes of convergence (annotated index)Cauchy principal valueFubinis theorem1/2 1/4 + 1/8 1/16 + 1/2 + 1/4 + 1/8 + 1/16 + 1.7 Notes[1] Megginson, Robert E. (1998), An introduction to Banach space theory, Graduate Texts in Mathematics 183, New York:Springer-Verlag, p. 20, ISBN 0-387-98431-3 (Theorem 1.3.9)[2] Dvoretzky, A.; Rogers, C. A. (1950), Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad.Sci. U. S. A. 36:192197.1.8 ReferencesWalter Rudin, Principles of Mathematical Analysis (McGraw-Hill: New York, 1964).Chapter 2Addition chainIn mathematics, an addition chain for computing a positive integer n can be given by a sequence of natural numbersv and a sequence of index pairs w such that each term in v is the sum of two previous terms, the indices of thoseterms being specied by w:v =(v0,...,vs), with v0 = 1 and vs = nfor each 0< i s holds: vi = vj + vk, with wi=(j,k) and 0 j,k i 1Often only v is given since it is easy to extract w from v, but sometimes w is not uniquely reconstructible. Anintroduction is given by Knuth.[1]2.1 ExamplesAs an example: v = (1,2,3,6,12,24,30,31) is an addition chain for 31 of length 7, since2 = 1 + 13 = 2 + 16 = 3 + 312 = 6 + 624 = 12 + 1230 = 24 + 631 = 30 + 1Addition chains can be used for addition-chain exponentiation: so for example we only need 7 multiplications tocalculate 531:52= 51 5153= 52 5156= 53 53512= 56 56524= 512 512530= 524 56531= 530 5162.2. METHODS FOR COMPUTING ADDITION CHAINS 72.2 Methods for computing addition chainsCalculating an addition chain of minimal length is not easy; a generalized version of the problem, in which one mustnd a chain that simultaneously forms each of a sequence of values, is NP-complete.[2] There is no known algorithmwhich can calculate a minimal addition chain for a given number with any guarantees of reasonable timing or smallmemory usage. However, several techniques to calculate relatively short chains exist. One very well known techniqueto calculate relatively short addition chains is the binary method, similar to exponentiation by squaring. Other well-known methods are the factor method and window method.[3]2.3 Chain lengthLet l(n) denote the smallest s so that there exists an addition chain of length s which computes n. It is known that [4]log2(n) + log2((n)) 2.13 l(n) log2(n) + log2(n)(1 +o(1))/ log2(log2(n))where (n) is Hamming weight of binary expansion of n.It is clear that l(2n) l(n)+1. Strict inequality is possible, as l(382) = l(191) = 11, observed by Knuth.[5] The rstinteger with l(2n) < l(n) is n = 375494703.[6]2.4 Brauer chainA Brauer chain or star addition chain is an addition chain in which one of the summands is always the previouschain: that is,for each k>0: ak = ak-1 + aj for some j < k.A Brauer number is one for which the Brauer chain is minimal.[5]Brauer proved thatl*(2n1) n 1 + l*(n)where l* is the length of the shortest star chain. For many values of n,and in particular for n 2500, they are equal:l(n) = l*(n). But Hansen showed that there are some values of n for which l(n) l*(n), such as n = 26106+ 23048+22032+ 22016+ 1 which has l*(n) = 6110, l(n) 6109.2.5 Scholz conjectureMain article: Scholz conjectureThe Scholz conjecture (sometimes called the ScholzBrauer or BrauerScholz conjecture), named after A. Scholz andAlfred T. Brauer), is a conjecture from 1937 stating thatl(2n 1) n 1 + l(n) .It is known to be true for Hansen numbers, a generalization of Brauer numbers; N. Clift checked by computer thatall n5784688 are Hansen (while 5784689 is not).[6] Clift further checked that is true with equality for n64.[5]8 CHAPTER 2. ADDITION CHAIN2.6 See alsoAddition chain exponentiationAddition-subtraction chainVectorial addition chainLucas chain2.7 References[1] D. E. Knuth, The Art of Computer Programming, Vol 2, Seminumerical Algorithms, Section 4.6.3, 3rd edition, 1997[2] Downey, Peter; Leong, Benton; Sethi, Ravi (1981). Computing sequences with addition chains. SIAM Journal onComputing 10 (3): 638646. doi:10.1137/0210047.. A number of other papers state that nding a single addition chain isNP-complete, citing this paper, but it does not claim or prove such a result.[3] Otto, Martin (2001), Brauer addition-subtraction chains (PDF), Diplomarbeit, University of Paderborn.[4] A. Schnhage A lower bound on the length of addition chains, Theoret. Comput. Sci. 1 (1975), 112.[5] Guy (2004) p.169[6] Clift, Neill Michael (2011). Calculating optimal addition chains (PDF). Computing 91 (3): 265284. doi:10.1007/s00607-010-0118-8.Brauer, Alfred (1939). On addition chains. Bulletin of the American Mathematical Society 45 (10): 736739.doi:10.1090/S0002-9904-1939-07068-7. ISSN 0002-9904. MR 0000245.Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-20860-7.OCLC 54611248. Zbl 1058.11001. Section C6.2.8 External linkshttp://wwwhomes.uni-bielefeld.de/achim/addition_chain.html"Sloanes A003313 : Length of shortest addition chain for n", The On-Line Encyclopedia of Integer Sequences.OEIS Foundation.F. Bergeron, J. Berstel. S. Brlek Ecient computation of addition chainsChapter 3Addition-chain exponentiationIn mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation bypositive integer powers that requires a minimal number of multiplications. It works by creating the shortest additionchain that generates the desired exponent. Each exponentiation in the chain can be evaluated by multiplying two ofthe earlier exponentiation results. More generally, addition-chain exponentiation may also refer to exponentiation bynon-minimal addition chains constructed by a variety of algorithms (since a shortest addition chain is very dicult tond).The shortest addition-chain algorithm requires no more multiplications than binary exponentiation and usually less.The rst example of where it does better is for a15, where the binary method needs six multiplications but a shortestaddition chain requires only ve:a15= a (a [a a2]2)2a15= a3([a3]2)2On the other hand, the determination of a shortest addition chain is hard: no ecient optimal methods are currentlyknown for arbitrary exponents, and the related problemof nding a shortest addition chain for a given set of exponentshas been proven NP-complete.[1] Even given a shortest chain, addition-chain exponentiation requires more memorythan the binary method, because it must potentially store many previous exponents from the chain. So in practice,shortest addition-chain exponentiation is primarily used for small xed exponents for which a shortest chain can beprecomputed and is not too large.There are also several methods to approximate a shortest addition chain, and which often require fewer multiplica-tions than binary exponentiation; binary exponentiation itself is a suboptimal addition-chain algorithm. The optimalalgorithm choice depends on the context (such as the relative cost of the multiplication and the number of times agiven exponent is re-used).[2]The problem of nding the shortest addition chain cannot be solved by dynamic programming, because it does notsatisfy the assumption of optimal substructure. That is, it is not sucient to decompose the power into smallerpowers, each of which is computed minimally, since the addition chains for the smaller powers may be related (toshare computations). For example, in the shortest addition chain for a15above, the subproblem for a6must becomputed as (a3)2since a3is re-used (as opposed to, say, a6= a2(a2)2, which also requires three multiplies).3.1 Addition-subtractionchain exponentiationIf both multiplication and division are allowed, then an addition-subtraction chain may be used to obtain even fewertotal multiplications+divisions (where subtraction corresponds to division). However, the slowness of division com-pared to multiplication makes this technique unattractive in general. For exponentiation to negative integer powers,on the other hand, since one division is required anyway, an addition-subtraction chain is often benecial. Onesuch example is a31, where computing 1/a31by a shortest addition chain for a31requires 7 multiplications and onedivision, whereas the shortest addition-subtraction chain requires 5 multiplications and one division:910 CHAPTER 3. ADDITION-CHAIN EXPONENTIATIONa31= a/((((a2)2)2)2)2For exponentiation on elliptic curves, the inverse of a point (x, y) is available at no cost, since it is simply (x, y), andtherefore addition-subtraction chains are optimal in this context even for positive integer exponents.[3]3.2 References[1] Downey, Peter; Leong, Benton; Sethi, Ravi (1981). Computing sequences with addition chains. SIAM Journal onComputing 10 (3): 638646. doi:10.1137/0210047.[2] Gordon, D. M. (1998). Asurvey of fast exponentiation methods (PDF). J. Algorithms 27: 129146. doi:10.1006/jagm.1997.0913.[3] Franois Morain and Jorge Olivos, "Speeding up the computations on an elliptic curve using addition-subtraction chains,RAIRO Informatique thoretique et application 24, pp. 531-543 (1990).Donald E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd edition, 4.6.3(Addison-Wesley: San Francisco, 1998).Daniel J. Bernstein, "Pippengers Algorithm, to be incorporated into authors High-speed cryptography book.(2002)Chapter 4Addition-subtraction chainAn addition-subtraction chain, a generalization of addition chains to include subtraction, is a sequence a0, a1, a2,a3, ... that satisesa0= 1,fork > 0, ak= ai aj some for 0 i, j< k.An addition-subtraction chain for n, of length L, is an addition-subtraction chain such that aL= n . That is, one canthereby compute n by L additions and/or subtractions. (Note that n need not be positive. In this case, one may alsoinclude a=0 in the sequence, so that n=1 can be obtained by a chain of length 1.)By denition, every addition chain is also an addition-subtraction chain, but not vice versa. Therefore, the lengthof the shortest addition-subtraction chain for n is bounded above by the length of the shortest addition chain for n.In general, however, the determination of a minimal addition-subtraction chain (like the problem of determining aminimum addition chain) is a dicult problem for which no ecient algorithms are currently known. The relatedproblemof nding an optimal addition sequence is NP-complete (Downey et al., 1981), but it is not known for certainwhether nding optimal addition or addition-subtraction chains is NP-hard.For example, one addition-subtraction chain is:a0= 1 , a1= 2 = 1 + 1 , a2= 4 = 2 + 2 , a3= 3 = 4 1 . Thisis not a minimal addition-subtraction chain for n=3, however, because we could instead have chosen a2= 3 = 2 +1. The smallest n for which an addition-subtraction chain is shorter than the minimal addition chain is n=31, whichcan be computed in only 6 additions (rather than 7 for the minimal addition chain):a0= 1,a1= 2 = 1+1,a2= 4 = 2+2,a3= 8 = 4+4,a4= 16 = 8+8,a5= 32 = 16+16,a6= 31 = 321.Likeanadditionchain, anaddition-subtractionchaincanbeusedforaddition-chainexponentiation: giventheaddition-subtraction chain of length L for n, the power xncan be computed by multiplying or dividing by x L times,where the subtractions correspond to divisions. This is potentially ecient in problems where division is an inexpen-sive operation, most notably for exponentiation on elliptic curves where division corresponds to a mere sign change(as proposed by Morain and Olivos, 1990).Some hardware multipliers multiply by n using an addition chain described by n in binary:n = 31 = 0 0 0 1 1 1 1 1 (binary).Other hardware multipliers multiply by n using an addition-subtraction chain described by n in Booth encoding:n = 31 = 0 0 1 0 0 0 0 1 (Booth encoding).4.1 ReferencesHugo Volger, Some results on addition/subtraction chains, Information Processing Letters 20, pp. 155160(1985).1112 CHAPTER 4. ADDITION-SUBTRACTION CHAINFranois Morain and Jorge Olivos, "Speeding up the computations on an elliptic curve using addition-subtractionchains, RAIRO Informatique thoretique et application 24, pp. 531543 (1990).Peter Downey, Benton Leong, and Ravi Sethi, Computing sequences with addition chains, SIAMJ. Computing10 (3), 638-646 (1981).Sequence A128998, length of minimum addition-subtraction chain, The On-Line Encyclopedia of IntegerSequences.Chapter 5Almost convergent sequenceA bounded real sequence (xn) is said to be almost convergent to L if each Banach limit assigns the same value L tothe sequence (xn) .Lorentz proved that (xn) is almost convergent if and only iflimpxn +. . . +xn+p1p= Luniformly in n .The above limit can be rewritten in detail as( > 0)(p0)(p > p0)(n)xn +. . . +xn+p1pL < .Almost convergence is studied in summability theory. It is an example of a summability method which cannot berepresented as a matrix method.5.1 ReferencesG. Bennett and N.J. Kalton: Consistency theorems for almost convergence. Trans. Amer. Math. Soc.,198:23-43, 1974.J. Boos: Classical and modern methods in summability. Oxford University Pres