MACLAURIN SERIES
how to represent certain types of functions as sums of power series
You might wonder why we would ever want to express a known function as a sum of infinitely many terms.
Integration. (Easy to integrate polynomials)
Finding limit
Finding a sum of a series (not only geometric, telescoping)
dxex2
20
1lim
x
xex
x
Example: xexf )(
0n
nn
x xce 55
44
33
2210 xcxcxcxcxcc
Maclaurin series ( center is 0 )
Example:
xxf sin)( Find Maclaurin series
MACLAURIN SERIES
xxf sin)(
xxf cos)()1(
xxf sin)()2(
xxf cos)()3(
xxf sin)()4(
0)0( f
1)0()1( f
0)0()2( f
1)0()3( f
0)0()4( f
!
)0()(
n
nfnc
Example:
xxf cos)( Find Maclaurin series
MACLAURIN SERIES
xxdx
dcossin
Example:
xxf
1
1)(
Find Maclaurin series
Example:
21
1)(x
xf
Find Maclaurin series
0
22
)(1
1
n
nxx
2by each replace xx
0
2)1(
n
nn x
Example:
xxf 1tan)(
Find Maclaurin series
integrate
0
)1(2
1 2
1tan
n
dxxn n
x
dxx
0
12)1(
12
nn
xn n
Example:
xxf
1
1)(
Find Maclaurin series
0
)(1
1
n
nxx
xx by each replace
0
)1(
n
nn x
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620
MACLAURIN SERIES
How to memorize them
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620
MACLAURIN SERIES
Denominator is n!
even, odd
Denominator is nodd
MACLAURIN SERIES
Maclaurin series ( center is 0 ) !
)0()(
n
nfnc
How to find a Maclaurin Series of a function
Use the formula Use the known functions
1) Replace each x2) Diff 3) integrate3) Find a product between two
TERM-081
MACLAURIN SERIES
TERM-091
MACLAURIN SERIES
TERM-101
MACLAURIN SERIES
TERM-082
)2cos(cos2
1
2
12 xx
MACLAURIN SERIES
TERM-102
MACLAURIN SERIES
TERM-091
MACLAURIN SERIES
TAYLOR AND MACLAURIN
Example:
0 !
1
n nFind the sum of the series
TERM-102
MACLAURIN SERIES
TERM-082
MACLAURIN SERIES
TERM-131
MACLAURIN SERIES
Leibniz’s formula:
Example: Find the sum
0
121
12)1()(tan
n
nn
n
xx
753)(tan
7531 xxx
xx
0 12
)1(
n
n
n
7
1
5
1
3
11
MACLAURIN SERIES
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620
MACLAURIN SERIES
Denominator is n!
even, odd
Denominator is nodd
MAC
Important Maclaurin Series and Their Radii of Convergence
Example:
)1ln()( xxf
Find Maclaurin series
MACLAURIN SERIES
TERM-122
MACLAURIN SERIES
TERM-082
MACLAURIN SERIES
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620
MACLAURIN SERIES
Denominator is n!
even, odd
Denominator is nodd