AP CALCULUS BC
Section Number:
9.9
LECTURE NOTES
Topics: Representations of Functions by Power Series
- Geometric Power Series
- Operations with Power Series (Adding)
MR. RECORD
Day: 25
Geometric Power Series
Goal: Investigate techniques for finding a power series that represents a given function.
Consider the function f ( x)  1 . The form of f closely resembles
1 x

 ar
n 0
n

a
, r  1.
1 r
Put another way, if you let a = 1 and r = x, a power series representation for

1
  xn
1  x n 0
 1  x  x 2  x3 
1 , centered at 0, is
1 x
Note: This series is represented on the interval (-1, 1)
, x  1.
To represent f on another interval, we must develop a completely different series.
Let’s try one centered at -1.
1
1
1/ 2
a



1  x 2  ( x  1) 1   x  1 / 2  1  r
which implies that a = ½ and r = (x+1)/2.
Example 1: Finding a Geometric Power Series Centered at 0.
4
Find a power series for f ( x) 
, centered at 0 and state its interval of convergence.
x2
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Example 1
Example 2: Finding a Geometric Power Series Centered at 1.
1
Find a power series for f ( x )  , centered at 1 and state its interval of convergence.
x
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Example 2
Operations with Power Series
The following properties will make it very easy to write more complicated power series.
OPERATIONS WITH POWER SERIES


Let f ( x)   an x n and g ( x)   bn x n .
n 0
n 0


n 0
n 0


1. f (kx)   an (kx) n   an k n x n
2. f ( x N )   an ( x N ) n   an x nN
n 0
n 0

3. f ( x)  g ( x)    an  bn  x n
n 0
Example 3: Adding Two Power Series.
Find a power series, centered at 0, for f ( x) 
3x  1
. Be sure to state its interval of
x2 1
convergence.
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Example 3
AP CALCULUS BC
Section Number:
9.9
LECTURE NOTES
Topics: Representations of Functions by Power Series
- Finding a Power Series by Integration
MR. RECORD
Day: 26
Example 4: Finding a Power Series by Integration.
a.
Find a power series, centered at 1, for f ( x)  ln x . Be sure to state its interval of convergence.
b.
Find a power series, centered at 0, for f ( x)  arctan x . Be sure to state its interval of convergence.
Example 5: Approximation using a Power Series

1
x 2 n1
Use the series for f ( x)  arctan x ,  (1)n
to approximate arctan using RN  0.001 .
,
3
2n  1
n 0