the comparison tests theorem suppose that and are series with positive terms, then (i) if is...
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The comparison tests Theorem Suppose that and are series with
positive terms, then
(i) If is convergent and for all n, then is also convergent.
(ii) If is divergent and for all n, then is also
divergent. Ex. Determine whether converges.
Sol. So the series converges.
na nb
nanb n na b
nanb n na b
1
1
2 1nn
1 1
2 1 2n n
The limit comparison test Theorem Suppose that and are series with
positive terms. Suppose
Then
(i) when c is a finite number and c>0, then either both series
converge or both diverge.
(ii) when c=0, then the convergence of implies the
convergence of
(iii) when then the divergence of implies the
divergence of
na nb
lim .n
nn
ac
b
.nanb
,c nb.na
Example Ex. Determine whether the following series converges.
Sol. (1) diverge. choose then
(2) diverge. take then
(3) converge for p>1 and diverge for take
then
21
1(2)
ln ( 1)n n
2
51
2 3(1)
5n
n n
n
1
(3) sin p
n n
1/ 21/nb n lim / 2n nn
a b
1/nb n lim /n nna b
1/ pnb n1p
lim / pn n
na b
Question Ex. Determine whether the series
converges or diverges.
Sol.
1ln
1
( 0)n
n
a a
1 1ln ln ln
ln
1an n
n aa a e
n
diverge for 0 a e
converge for a e
Alternating series An alternating series is a series whose terms are alternati
vely positive and negative. For example,
The n-th term of an alternating series is of the form
where is a positive number.
1
1
1 1 1 ( 1)1
2 3 4
n
n n
nb
1( 1) or ( 1)n nn n n na b a b
The alternating series test Theorem If the alternating series
satisfies (i) for all n (ii)
Then the alternating series is convergent.
Ex. The alternating harmonic series
is convergent.
11 2 3 4 5 6
1
( 1) ( 0)nn n
n
b b b b b b b b
1
1
( 1)n
n n
1n nb b lim 0nn
b
Example Ex. Determine whether the following series converges.
Sol. (1) converge (2) converge
Question.
1 1 2
31 1
( 1) ( 1)(1) ( 0) (2)
1
n n
n n
n
n n
1
1
( 1)
4 1
n
n
n
n
Absolute convergence A series is called absolutely convergent if the series
of absolute values is convergent.
For example, the series is absolutely convergent
while the alternating harmonic series is not. A series is called conditionally convergent if it is
convergent but not absolutely convergent. Theorem. If a series is absolutely convergent, then it is
convergent.
na| |na
1
3/ 21
( 1)n
n n
na
Example Ex. Determine whether the following series is convergent.
Sol. (1) absolutely convergent
(2) conditionally convergent
21 1
sin ( 1)(1) (2)
ln(1 )
n
n n
n
n n
The ratio test The ratio test
(1) If then is absolutely convergent.
(2) If or then diverges.
(3) If the ratio test is inconclusive: that is, no
conclusion can be drawn about the convergence of
1lim 1,n
nn
aL
a
1n
n
a
1lim 1n
nn
aL
a
1lim n
nn
a
a
1n
n
a
1lim 1,n
nn
a
a
1n
n
a
Example Ex. Test the convergence of the series
Sol. (1) convergent
(2) convergent for divergent for
1 1
!(1) (2)
!
n n
nn n
a a n
n n
;a e a e1
11
( 1)! !/
( 1) (1 1/ )
n nn
n n nn
a a n a n a a
a n n n e
1 limn n nna e a a a
0
The root test The root test
(1) If then is absolutely convergent.
(2) If or then diverges.
(3) If the root test is inconclusive.
lim | | 1,nn
na L
1n
n
a
lim | | 1nn
na L
lim | |n
nn
a
1
nn
a
lim | | 1,nn
na
Example Ex. Test the convergence of the series
Sol.
convergent for divergent for
1
( 0)1
nn
na
an
1;a 0 1a
1lim lim
1
nn
nn n
na
aan
1 ( )1
(1 )n
n
na a n
n
Rearrangements If we rearrange the order of the term in a finite sum, then of
course the value of the sum remains unchanged. But this is not the case for an infinite series.
By a rearrangement of an infinite series we mean a series obtained by simply changing the order of the terms.
It turns out that: if is an absolutely convergent series with sum , then any rearrangement of has the same sum .
However, any conditionally convergent series can be rearranged to give a different sum.
s
na
nas
na
Example Ex. Consider the alternating harmonic series
Multiplying this series by we get
or
Adding these two series, we obtain
1/ 2,
1 1 1 1 11 ln 2.
2 3 4 5 6
1 1 1 1 1 1 1ln 2.
2 4 6 8 10 12 2
1 1 1 1 1 31 ln 2.
3 2 5 7 4 2
1 1 1 1 10 0 0 0 ln 2.
2 4 6 8 2
Strategy for testing series If we can see at a glance that then divergence
If a series is similar to a p-series, such as an algebraic form, or a form containing factorial, then use comparison test.
For an alternating series, use alternating series test.3
4( 1)
1n
n
na
n
1 10
2 1 2n
na
n
3
3 2 3/ 2
1 1~
3 4 2 3n
na
n n n
lim 0nn
a
Strategy for testing series If n-th powers appear in the series, use root test.
If f decreasing and positive, use integral test.
Sol. (1) diverge (2) converge (3) diverge (4) converge
( ),na f n
2nna ne
1
(ln )(ln ln )nan n n
ln
1 ( 1) ln 2 ! 1(1) tan (2) (3) (4)
( 2)! (ln )
n n
n
n n
n n nn
Power series A power series is a series of the form
where x is a variable and are constants called coefficients
of series. For each fixed x, the power series is a usual series. We can
test for convergence or divergence. A power series may converge for some values of x and
diverge for other values of x. So the sum of the series is a function
2 30 1 2 3
0
nn
n
c x c c x c x c x
2 30 1 2 3( )s x c c x c x c x
nc
Power series For example, the power series
converges to when
More generally, A series of the form
is called a power series in (x-a) or a power series centered
at a or a power series about a.
20 1 2
0
( ) ( ) ( )nn
n
c x a c c x a c x a
2 3
0
1n
n
x x x x
1( )
1s x
x
1 1.x
Example Ex. For what values of x is the power series
convergent? Sol. By ratio test,
the power series diverges for all and only converges
when x=0.
0
! n
n
n x
11 ( 1)!
lim lim lim( 1) | |!
nn
nn n nn
a n xn x
a n x
0,x
Homework 24 Section 11.4: 24, 31, 32, 42, 46
Section 11.5: 14, 34
Section 11.6: 5, 13, 23
Section 11.7: 7, 8, 10, 15, 36