AP CALCULUS BC
Section Number:
LECTURE NOTES
MR. RECORD
Day: 21
Topics: Power Series
- Power Series
- Radius and Interval of Convergence
9.8
Power Series
Goal: Show how an infinite polynomial functions can be used as an EXACT expression for other elementary
functions.
For example: The function f ( x)  e x can be represented exactly by
x 2 x3
xn
We call this a power series.
   
2! 3!
n!
For each real number, x, it can be shown that the infinite series on the right converges to the value e x .
ex  1  x 
DEFINITION OF POWER SERIES
If x is a variable, then an infinite series of the form

a x
n 0
n
n
 a0  a1 x  a2 x 2  a3 x 3 
an x n 
is called a power series. More generally, an infinite series of the form

 a ( x  c)
n 0
n
n
 a0  a1 ( x  c)  a2 ( x  c) 2  a3 ( x  c )3 
an ( x  c ) n 
is called a power series centered at c, where c is a constant.
Example 1: Power Series.
State the value of c where each power series is centered.

xn
a. 
b. 1  ( x  1)  ( x  1)2  ( x  1)3 
n
!
n 0

c.
1
 n ( x  1)
n
n 0
Radius and Interval of Convergence
Think of a power series as a function of x like

f ( x)   an ( x  c) n
n 0
where the domain of f is the set of all x for which the power series
converges.
Determining the domain of a power series is the primary
concerning this section.
Note: A power series converges at its center c because

f (c)   an (c  c) n  a0 (1)  0  0  0 
n 0
So, c always lies in the domain of f
0
 a0
The 3 Types of Convergence of a Power Series
THEOREM 9.20 CONVERGENCE OF A POWER SERIES
For a power series centered at c, precisely one of the following is true.
1. The series converges only at c.
2. There exists a real number R > 0 such that the series converges absolutely for
x  c  R, and diverges for x  c  R.
3. The series converges absolute for all x.
The number R is called the radius of convergence of the power series. If the series converges
only at c, the radius of convergence is R = 0, and if the series converges for all x, the radius of
convergence is R = ∞.
The set of all values of x for which the power series converges is the interval of convergence of
the power series.
Example 2: Finding the Radius of Convergence.
Find the radius of convergence for each power series.

a.
 n! xn
n 0

b.
 3( x  2)
n
n0
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Example 2a
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Example 2b
(1)n x 2 n1

n 0 (2n  1)!

c.
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Example 2c
AP CALCULUS BC
Section Number:
9.8
LECTURE NOTES
Topics: Power Series
- Endpoint Convergence
- Differentiation and Integration of Power
Series
MR. RECORD
Day: 22
Endpoint Convergence
Note that for a power series whose radius of convergence is a finite number R, the theorem above says
nothing about the convergence at the endpoints of the interval of convergence. Each endpoint must be tested
separately for convergence or divergence. Therefore, the interval can take any one of the following six forms:
Example 3: Finding the Interval of Convergence.
Find the interval of convergence for each power series.

xn
a. 
n 1 n
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Example 3a
(1)n ( x  1)n

2n
n 0

b.

c.
xn

2
n 1 n
Differentiation and Integration of Power Series
All of the “founding fathers” of calculus, Newton, Leibniz, Euler, Lagrange and the Bernoullis used power
series extensively in their individual development of the subject.
Because of this, one can only wonder are power series continuous? Differentiable? Integrable?
THEOREM 9.21 PROPERTIES OF FUNCTIONS DEFINED BY POWER SERIES
If the function given by

f ( x)   an ( x  c) n
n 0
 a0  a1 ( x  c)  a2 ( x  c) 2  a3 ( x  c)3  an ( x  c) n 
has radius of convergence of R > 0, then, on the interval (c – R, c + R), f is differentiable (and
therefore continuous). Moreover, the derivative and antiderivative of f are as follows:

1.
f ( x)   nan ( x  c) n 1
n 1
 a1  2a2 ( x  c)  3a3 ( x  c) 2 

2.

f ( x) dx  C   an
n 0
( x  c)n 1
n 1
 C  a0 ( x  c)  a1
( x  c) 2
( x  c)3
 a2

2
3
The radius of convergence of the series obtained by differentiating or integrating a power series
is the same as that of the original power series. The interval of convergence, however, may
differ as a result of the behavior at the endpoints.
Activity:

1. Given f ( x)  
n 0
xn
, write out an expression that includes the first five terms of the series.
n!
2. Find f ( x ) by differentiating the terms in the series above.
3. What do you notice about f ( x) and f ( x) ?
Do you recognize this function?
Example 4: Intervals of Convergence for f(x), f'(x), and ʃf(x) dx.

xn
x 2 x3
Consider the function given by f ( x)    x    . .
2 3
n 1 n
Find the interval of convergence for each of the following.
a. f ( x)
b. f ( x )
c.
 f ( x) dx