Calc 2 Lecture Notes
Section 8.6
Page 1 of 5
Section 8.6: Power Series
Big idea: Series involving a variable raised to a power are called power series. They are
extremely useful as an alternative way to represent any function, since their computation
involves only addition, subtraction and multiplication.
Big skill:. You should be able to compute the radius of convergence of a given power series, to
differentiate and integrate a power series term by term, and to obtain power series
representations of a few basic functions related to geometric power series.
Power Series Definition
A power series in powers of (x – c) is any series of the form

b  x  c
k 0
k
k
 b0  b1  x  c   b2  x  c   b3  x  c  
2
3
,
where the constants bk are called the coefficients of the series.

One thing to notice is that a power series looks a lot like a geometric series:
 a r 
k 0
k

a
,
1 r
for |r| < 1.
So, right off the bat we can use a power series to represent a function. Let’s choose to let bk = 1
and c = 2:

1  x  2 
k
 1   x  2   x  2   x  2 
2
3
k 0
This is now exactly in the form of a power series, with r = x – 2. So, the series will converge as

1
1
k

long as |x – 2| < 1 (or 1 < x < 3). The series converges to: 1  x  2  
, so
1   x  2 3  x
k 0

1
k
   x  2  . The graph below
3  x k 0
shows the function and the first three partial sum functions S1(x) = 1 + (x – 2),
S2(x) = 1 + (x – 2) + (x – 2)2, and S3(x) = 1 + (x – 2) + (x – 2)2 + (x – 2)3.
we have another way to represent this function… f  x  
Calc 2 Lecture Notes
Section 8.6
Page 2 of 5
Determining the values of x for which a power series converges
We just saw an example of creating our own power series so that it matched the geometric series,
and then we used the formula for the sum of a geometric series to find the x values for which the
power series converges. We found that it converged for |x – 2| < 1 
x  (1, 3) = (2-1, 2+1) = (c – r, c + r). The constant r is called the radius of convergence, and
the interval of x values (c – r, c + r) over which the series converges is called the interval of
convergence. In general, we can say this about the convergence of a power series:
Theorem 6.1: Convergence of a Power Series

Given any power series  bk  x  c  , there are exactly three possibilities:
k
k 0
The series converges absolutely for all x  (-, ), and the radius of convergence is
r = .
(ii). The series converges only for x = c, and the radius of convergence is r = 0.
(iii). The series converges absolutely for x  (c – r, c + r), for some radius of convergence r
with 0 < r < , and diverges for |x – r| > c. The endpoints of the interval need to be evaluated
separately.
(i).
Now we will look at what to do when someone else gives us a power series, and asks for what
range of x values it converges.
Primary Tool to Test for Convergence of a Power Series: The Ratio Test
Recall: Theorem 5.2: The Ratio Test

Given
a
k 1
(iv).
(v).
(vi).
k
, with ak  0 for all k, suppose that lim
k 
ak 1
 L . Then:
ak
if L < 1, the series converges absolutely.
if L > 1 (or L = ), the series diverges.
if L =1, no conclusion can be made.
 Strategy: Solve for x: lim
k 
ak 1
1
ak
Practice:

Determine the radius of convergence of the power series
k
3
k 0
k 1
xk .
Calc 2 Lecture Notes
Section 8.6
Page 3 of 5

Determine the radius of convergence of the power series
10k
 k !  x  1
k
.
k 0

Determine the radius of convergence of the power series
k 0

Determine the radius of convergence of the power series
1
 k4
k
xk .
 k ! x  5
k
.
k 0
Calculus Operations on a Power Series
If we define a function f(x) on the interval of convergence of a power series:

f  x    bk  x  c  , then we can obtain the derivative and integral of the function by
k
k 0
differentiating or integrating each term of the power series.
d
2
3
f  x 
b0  b1  x  c   b2  x  c   b3  x  c  
dx


 b1  2b2  x  c   3b3  x  c  
2

  k  bk  x  c 
k 1
k 1
And
 f  x  dx    b
0
 b1  x  c   b2  x  c   b3  x  c  
2
3
 dx
1
1
1
2
3
4
 b0 x  b1  x  c   b2  x  c   b3  x  c    K I
2
3
4

1
k 1

 bk  x  c   K I
k 1 k  1
Note: These formulas only apply to power series, and not to any series in general.
Calc 2 Lecture Notes
Section 8.6
Page 4 of 5
Practice:

Show that the power series
  1
k
x k is a power series representation of f  x  
k 0

Use the power series
  1
k
k 0

Use the power series
  1
k 0
k
x k to find a power series representation of f  x  
1
.
1 x
1
1  x 
2
.
x k to find a power series representation of f  x   ln 1  x  .
Calc 2 Lecture Notes
Section 8.6
Use the geometric to find a power series representation of f  x  
Find a power series representation of f  x   tan 1  x  .
Use the previous result to calculate an approximate value of .
Page 5 of 5
1
.
1  x2