Section 8.7: Power Series
Definition


k 0
ak ( x  c) k is a power series centered at c
a0  a1 ( x  c)  a2 ( x  c) 2  a3 ( x  c)3  ...
If c = 0,
a0  a1 x  a2 x  a3 x  ...
2
3
IMPORTANT:
A power series is a function.
Its value and whether or not it converges
depends on which x you plug in.
Think of them as “Long Polynomials”

k
x

k 0
If you plug in a number for x,
you get a geometric series.
It converges if |x|<1.
Remember, a power series is a function.

x
k 0
k
1

1 x

2
k 0
lim
k 
3
k
k
( x  5) k Find all values of x at which the power series converges.
3 ( x  5) k 1
2k 1 (k  1)
2k k
3 ( x  5) k
k | x 5|
 lim
k 
2 (k  1)

| x 5|
2
| x 5|
 1
By ratio test the series converges absolutely when
2
| x 5|  2
| x 5|  2
If x-5 is positive:
If x-5 is negative:
x 5  2
| x 5|  2
x  7
x 5   2
x  3
| x 5|  2
Let’s try geometry instead of algebra.
|x-5| is really just the distance from x to 5
2
3
2
5
7
The Radius of Convergence is 2.

3
By ratio test  2k k
( x  5)k
k 0
converges absolutely when 3 < x < 7.
But we must still check endpoints
because the ratio test tells us nothing at x = 3 and x = 7.
x7

2
k 0

x3
2
k 0
3
k
3
k
k
k
(7  5)
(3  5)
k


2
k 0
k


2
k 0
3
k
3
k
k
k
2

k


k 0
(2)
k

3
k


k 0
Diverges
3
(1) k Converges
k
Radius of convergence R = 2
Interval of convergence [3, 7)
 k  x  2
k
lim
k 
k
k
| k k ( x  2) k |
Try the Root Test this time.
 lim k | x  2 |  
k 
So the series converges only at x = 2
Radius of convergence is 0
Unless x=2
xk
 k!
lim
k 
x k 1
k!
(k  1)! x k
| x|
 0 No matter what x is!
 lim
k  k  1
So the series converges no matter what x is.
Radius of convergence = 
Interval of convergence= (, )
Theorem
k
a
(
x

c
)
 k
For a power series
exactly one of these
possibilities occurs:
1. It converges only at x = a
2. It converges at every number x
3. There is a number R > 0 so that
a) It converges absolutely if |x-a|<R
b) It diverges if |x-a|>R
Suppose
k
a
(
x

3)
 k
converges at x = 8
Does it converge at x = 6 ?
Yes !
Does it converge at x = -1 ?
Yes !
Does it converge at x = -5 ?
Maybe ?
Does it converge at x = 9?
Maybe ?
-5
3
8
The Calculus of Power Series
If a power series is a function,
can we integrate or differentiate it?
How much like a polynomial is it?
Theorem
d
dx


k 0

 a
k 0
k
ak ( x  a ) k
( x  a)



k 1

k
dx  
k 0
k ak ( x  a ) k 1
ak
( x  a ) k 1  C
k 1
This is valid on the interval of convergence of the original series.
d
dx


k 0
1 k
x 
k!


k 1
1
k x k 1
k!
d 
1 2
1 3
1 4

x  ... 
1  x  x  x 
dx 
2
6
24

1 2
1 3
0  1  x  x  x  ...
2
3

1 k
So what is  x ??
k 0 k !
It must be
x
e
!!
d
dx


k 0

 
k 0
3k 1
( x  4) k
(2k )!

3k 1
k
( x  4) dx 
(2k )!


k 0


k 0
3k 1
k 1
k ( x  4)
(2k )!
3k 1
1
k 1
( x  4)
C
(2k )! k  1