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(0
n
n
n
n
n
n
xn
xn
xxxx
xxxx
seriesrequiredThe
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)(0
:
n
n
n
n
n
n
xn
xn
xxxx
xxxx
seriesrequiredThe
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
xxx
seriesrequiredThe
2
0
642
642
)2/(!2
)1(
)2/(!6
1)2/(
!4
1)2/(
!2
11
)2/(!6
10)2/(
!4
10)2/(
!2
101
:
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
100
:
n
n
n
n
n
n
xn
xn
xxxx
xxxx
seriesrequiredThe
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
x
isxxforseriesMaclaurinThe
Hence
xn
isxforseriesMaclaurinThe
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
xn
xxxxx
seriesrequiredThe
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
2
232222
0
2
32
0
!
)1(
!
)(
:
)(
!
)(
!3
)(
!2
)()(1
!
)(
:
!!3!21
!
:
2
n
nn
n
n
n
n
n
x
n
n
n
x
n
x
n
x
thatNotice
xbyxreplaceWe
n
xxxx
n
x
iseforseriesMaclaurinTheHence
n
xxxx
n
x
iseforseriesMaclaurinThe
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
xxxxx
seriesrequiredThe
)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(
:
1
1
5432
5432
I-1Taylor Series for f(x) = cos2x about x = 5π
nf(n)(x)f(n)(5π)an=f(n)(5π) / n!an (x- 5π(n
0cos2x11/0!1/0! (x-5π)0=1
1-2 sin2x000
2-22 cos2x- 22- 22/ 2!- 22/ 2! (x-5π)2
323 sin2x000
424 cos2x2424/ 4!24/ 4! (x-5π)4
5-25 sin2x000
Taylor Series for cos2x about 5π
22
1
11
2
0
42
22
)5()!22(
2)1(
)5()!2(
2)1(
)5(!4
2)5(
!2
21
:
n
n
nn
n
n
nn
xn
Or
xn
xx
seriesrequiredThe
I-2Taylor Series for f(x) = cosx about x = 3π/4
nf(n)(x)f(n)(3π/4)an = f(n)(3π/4) / n!an (x- 3π/4 (n
0cosx- 1/√2- 1/√2 / 0!- 1/√2 / 0!(x-3π/4)0
1- sinx- 1/√2-1/√2 / 1!- 1/√2 / 1!(x-3π/4)1
2- cosx- (-1/√2)
= 1/√2
1/√2 / 2!1/√2 / 2!(x-3π/4)2
3sinx1/√21/√2 / 3!1/√2 / 3!(x-3π/4)3
4cosx- 1/√2- 1/√2 / 4!- 1/√2 / 4!(x-3π/4)4
5- sinx- 1/√2- 1/√2 / 5!- 1/√2 / 5!(x-3π/4)5
Taylor Series for cosx about 3π/4
0
1212
21
0
1212
0
21
53
420
54
320
)4
3(
)!12(
1
2
)1()
4
3(
!2
1
2
)1(
)4
3(
)!12(
1
2
)1( )
4
3(
!2
1
2
)1(
)4
3(
!5
1
2
1)
4
3(
!3
1
2
1)
4
3(
!1
1
2
1
)4
3(
!4
1
2
1)
4
3(
!2
1
2
1)
4
3(
!0
1
2
1
)4
3(
!5
1
2
1)
4
3(
!4
1
2
1
)4
3(
!3
1
2
1)
4
3(
!2
1
2
1)
4
3(
!1
1
2
1)
4
3(
!0
1
2
1
:
n
nn
nn
n
nn
n
nn
xn
xn
xn
xn
xxx
xxx
xx
xxxx
SeriesTaylorrequiredThe
I.3Taylor Series for f(x) = cosx about x = π/6
nf(n)(x)f(n)(π/6)an=f(n)(π/6) / n!an (x- π/6(n
0cosx√3/2√3/2 / 0!√3/2 / 0!(x-π/6)0
1- sinx- 1/2- ½ / 1!- 1/√2 / 1!(x-π/6)1
2- cosx- √3/2- √3/2 / 2!1/√2 / 2!(x-π/6)2
3sinx1/2½ / 3!1/√2 / 3!(x-π/6)3
4cosx- √3/2- √3/2 / 4!- 1/√2 / 4!(x-π/6)4
5- sinx- 1/2- ½ / 5!- 1/√2 / 5!(x-π/6)5
Taylor Series for cosx about π/6
0
121221
0
1212
0
21
53
420
54
320
)6
()!12(
1
2
1)1()
6(
!2
1
2
3)1(
)6
()!12(
1
2
1)1( )
6(
!2
1
2
3)1(
)6
(!5
1
2
1)
6(
!3
1
2
1)
6(
!1
1
2
1
)6
(!4
1
2
3)
6(
!2
1
2
3)
6(
!0
1
2
3
)6
(!5
1
2
1)
6(
!4
1
2
3
)6
(!3
1
2
1)
6(
!2
1
2
3)
6(
!1
1
2
1)
6(
!0
1
2
3
:
n
nnnn
n
nn
n
nn
xn
xn
xn
xn
xxx
xxx
xx
xxxx
SeriesTaylorrequiredThe
I.4Taylor Series for f(x) = e3x about x = 5
nf(n)(x)f(n)(5)an=f(n)(π) / n!an (x- 5(n
0e3xe15 e15/ 0!e15/ 0! (x-5)0
13e3x3 e153 e15 / 1!3 e15 / 1! (x-5)1
232e3x32 e1532 e15/ 2!32 e15/ 2! / 1! (x-5)2
333e3x33 e1533 e15 / 3!33 e15/ 2! / 1! (x-5)3
434e3x34 e1534 e15/ 4!34e15/ 2! / 1! (x-5)4
535e3x35 e1535 e15/ 5!35 e15/ 2! / 1! (x-5)5
0
15
5155
4154
2153
2152
1515
)5(!
3
)5(!5
3)5(
!4
3
)5(!3
3)5(
!2
3)5(3
:
n
nn
xen
xexe
xexexee
seriesrequiredThe
Homework
5)()5(
6sin)()4(
4
3cos)()3(
2
3cos)()2(
52cos)()1(
.
3
cexf
cxxf
cxxf
cxxf
cxxf
cxaboutfforseriesTaylortheFindI
x
1)1(cos)()4(
1)1sinh()()4(6
7sin)()3(
1cosh)()2(3
2cos)()1(
.
cxshxf
cxxf
cxxf
cxxf
cxxf
cxaboutfforseriesTaylortheFindII
xxf
exf
xxf
xxf
xxxf
xxf
xxf
xxf
xxf
xxf
fforseriesMaclaurintheFindIII
x
5cos2)()10(
2)()9(
sin)()8(
)1()()7(
cos)()6(
sinh)()5(
cosh)()4(
cos)()3(
)1ln()()2(
)1ln()()1(
.
5
3
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