taylor series. theorem definition the series is called the taylor series of f about c (centered at...
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Taylor Series
Theorem
0
)(
)(
0
)(!
)()(
,
:!
)(
)inf(
)()('
n
nn
n
n
n
nn
cxn
cfxf
thencaboutseriespowerahasfIf
wordsotherInn
cfaThen
esconvergencofradiusiniteorpositiveawith
cxaxfifsThat
cxabouttionrepresentaseriespowerahasfIf
Definition
The series
is called the Taylor series of f about c (centered at c)
0
)(
)(!
)(
n
nn
cxn
cf
Definition
The series
is called the Maclaurin series of f about c (centered at c)
Thus a Maclaurin series is a Taylor series centered at 0
0
)(
!
)0(
n
nn
xn
f
Examples I
Example (1)Taylor Series for f(x) = sinx about x = 2π
nf(n)(x)f(n)(2π)an=f(n)(2π) / n!an (x- 2π(n
0sinx000
1cosx11/1!(1/1!)(x- 2π(1
2-sinx000
3-cosx-1-1 / 3!(-1 / 3!)(x- 2π( 3
4sinx000
5cosx11/ 5!) 1/ 5!)(x- 2π(5
Taylor Series for sinx about 2π
12
0
12
1
1
753
753
)2()!12(
)1(
)2()!12(
)1(
)2(!7
1)2(
!5
1)2(
!3
1)2(
)2(!7
10)2(
!5
10)2(
!3
10)2(0sin
,
n
n
n
n
n
n
xn
xn
xxxx
xxxxx
haveWe
Example (2)Taylor Series for f(x) = sinx about x = π
nf(n)(x)f(n)(π)an=f(n)(π) / n!an (x- π(n
0sinx000
1cosx-1-1/1!(-1/1!)(x- π(1
2-sinx000
3-cosx11 / 3!(1 / 3!)(x- π( 3
4sinx000
5cosx-11-/ 5!1-)/ 5!)(x- π(5
Taylor Series for sinx about π
12
0
1
12
1
753
753
)()!12(
)1(
)()!12(
)1(
)(!7
1)(
!5
1)(
!3
1)(
)(!7
10)(
!5
10)(
!3
10)(0sin
,
n
n
n
n
n
n
xn
xn
xxxx
xxxxx
haveWe
Example (3)Taylor Series for f(x) = sinx about x = π/2
nf(n)(x)f(n)(π/2)an=f(n)(π/2) / n!an (x- π/2(n
0sinx11 / 0!(x- π/2(0=1
1cosx000
2-sinx-1-1 / 2!(-1 / 2!)(x- π/2(2
3-cosx000
4sinx11 / 4!(1 / 4!)(x- π/2(4
5cosx000
Taylor Series for sinx about π/2
n
n
n
xn
xxx
xxxx
haveWe
2
0
642
642
)2/(!2
)1(
)2/(!6
1)2/(
!4
1)2/(
!2
11
)2/(!6
10)2/(
!4
10)2/(
!2
101sin
,
Example (4)Maclaurin Series for f(x) = sinx
nf(n)(x)f(n)(0)an=f(n)(0) / n!an xn
0sinx000
1cosx11/1!(1/1!) x
2-sinx000
3-cosx-1-1 / 3!(-1 / 3! ) x3
4sinx000
5cosx11/ 5!(1 / 5! ) x5
Maclaurin Series for sinx
12
0
12
1
1
753
753
)!12(
)1(
)!12(
)1(
!7
1
!5
1
!3
1!7
10
!5
10
!3
100sin
,
n
n
n
n
n
n
xn
xn
xxxx
xxxxx
haveWe
Example(5)The Maclaurin Series for f(x) = x sinx
22
0
12
0
12
0
)!12(
)1(
)!12(
)1(sin
)!12(
)1(sin
,
n
n
n
n
n
n
n
n
n
xn
xn
xxx
Hence
xn
x
haveWe
Approximating sin(2○)
034899.0)90(!9
1)
90(!7
1)
90(!5
1)
90(!3
1
90)
90sin(
;,
)90(!9
1)
90(!7
1)
90(!5
1)
90(!3
1
90)
90sin(
90360
222
,
9753
9753
getweonlytermsfivefirstthegConsiderin
numberthetoscorrespond
haveWe
Examples II
Example (1)Maclaurin Series for f(x) = ex
nf(n)(x)f(n)(0)an=f(n)(0) / n!an xn
0ex11 /0!1
1ex11 /1!(1 / 1!) x
2ex11 /2!(1 / 2!) x2
3ex11 / 3!(1 / 3! ) x3
4ex11 /4!(1 / 4!) x4
5ex11/ 5!(1 / 5! ) x5
Maclaurin Series for ex
n
n
x
xn
xxxxxe
haveWe
0
5432
!
1!5
1
!4
1
!3
1
!2
11
,
Example (2)
Find a power series for the function
g(x) =2xe
0
2
0
2
2642
0
32
!
)1(
!
)(
!
)(
!3!21
!
!!3!21
2
n
nn
n
n
nx
n
n
nx
n
x
n
x
n
xxxxe
n
x
n
xxxxe
Example (3)TaylorSeries for f(x) = lnx about x=1
f(n)(x)f(n)(x)f(n)(1)an=f(n)(1) / n!an xn
0lnx000
1x-11=0!1(x -1)
2-x-2-1!-1!/2!(-1/2) (x -1)2
3(-1)(-2)x-32!2!/3!(1/3 ) (x -1)3
4(-1)(-2)(-3)x-4-3!-3!/4!(-1/4 (x -1)4
5(-1)(-2)(-3)(-4)x-54!4!/5!(1/5 ) (x -1)5
Taylor Series for lnx about x = 1
n
n
n
xn
xxxxx
xxxxxx
haveWe
)1()1(
)1(5
1)1(
4
1)1(
3
1)1(
2
1)1(
)1(!5
!4)1(
!4
!3)1(
!3
!2)1(
!2
!1)1(ln
,
1
1
5432
5432
Homework
1)()3(
2
3cos)()2(
cos)()1(
.
cexf
cxxf
cxxf
cxaboutfforseriesTaylortheFindI
x
xxf
exf
xxf
xxf
xxxf
xxf
xxf
xxf
xxf
xxf
fforseriesMaclaurintheFindII
x
cos)()10(
)()9(
sin)()8(
)1()()7(
cos)()6(
sinh)()5(
cosh)()4(
cos)()3(
)1ln()()2(
)1ln()()1(
.
5
3