ODE Lecture Notes
Section 5.1
Page 1 of 6
Section 5.1: Review of Power Series
Big Idea: In chapter 5, we will derive power series solutions to second-order differential
equations. Thus, some review is in order.
Big Skill: You should remember your calc 2 skills related to power series.
Review:

1. Definition of convergence of a power series: A power series
a x  x 
n 0
m
n
n
0
is said to
converge at a point x if lim  an  x  x0  exists for that x.
m 
n
n 0

2. Definition of absolute convergence of a power series: A power series
a x  x 
n 0

said to converge absolutely at a point x if
 a x  x 
n 0
n
0
n
n
0
n
is
converges.
a. Absolute convergence implies convergence…
3. Ratio test: If, for a fixed value of x, lim
n 
an 1  x  x0 
an  x  x0 
n 1
n
 x  x0 lim
n 
an 1
 x  x0 L , then
an
the power series converges absolutely at that value of x if x  x0 L  1, and diverges if
x  x0 L  1. If x  x0 L  1, then the test is inconclusive.
Practice:

n
Find the values of x for which  n x n converges.
n0 2
ODE Lecture Notes

4. If
a x  x 
n 0
n
Section 5.1
Page 2 of 6
converges at x = x1, it converges absolutely for x  x0  x1  x0 , and if
n
0
it diverges at x = x1, it diverges for x  x0  x1  x0 .
5. Radius and interval of convergence: The radius of convergence  is a nonnegative

number such that
a x  x 
n 0
n
0
n
converges absolutely for x  x0   and diverges for
x  x0   .
a. Series that converge only when x = x0 are said to have  = 0.
b. Series that converge for all x are said to have  = .
c. If  > 0, then the interval of convergence of the series is x  x0   .
Practice:
Find the radius of convergence of

Given that

 x  1
n 0
n 2n


n
.
 an  x  x0  and  bn  x  x0  converge for x  x0   …
n
n 0
n
n 0

6. Sum of series:


 a  x  x   b  x  x    a
n 0
n
n
0
n 0
n
n
0
n 0
n
 bn  x  x0 
n
ODE Lecture Notes
Section 5.1
Page 3 of 6
7. Product and Quotient of series:


n
n
a
x

x


0
 n
   bn  x  x0  
 n 0
  n 0

2
3
2
3
  a0  a1  x  x0   a2  x  x0   a3  x  x0   ... b0  b1  x  x0   b2  x  x0   b3  x  x0   ...





 a0 b0  b1  x  x0   b2  x  x0   b3  x  x0   ...
2
3


 a1  x  x0  b0  b1  x  x0   b2  x  x0   b3  x  x0   ...
2

a  x  x  b  b  x  x   b  x  x 
3


 b  x  x   ...
 a2  x  x0  b0  b1  x  x0   b2  x  x0   b3  x  x0   ...
2
3
3
0
0
1
0
2
0
2
2
3
3
3
0
...
 a0b0   a0b1  a1b0  x  x0    a0b2  a1b1  a2b0  x  x0    a0b3  a1b2  a2b1  a3b0  x  x0   ...
2
3

 n

n
    ak bn  k   x  x0 
n 0  k 0

To do a quotient, can write as a multiplication and equate terms:

a x  x 
n 0

n
b  x  x 
n 0
n
n
0
n




n
n
n
n
  d n  x  x0     d n  x  x0     bn  x  x0     an  x  x0 
n 0
 n 0
  n 0
 n 0
0
OR can do long division…
Practice:
Find the first few terms of the tangent function using the series for the sine and cosine functions.
ODE Lecture Notes
Section 5.1
Page 4 of 6
ODE Lecture Notes
Section 5.1
Page 5 of 6
8. Derivatives of a series:

d 
n
n 1
a
x

x

nan  x  x0 


n
0 

dx  n 0
 n 1
2


d 
n
n2
a
x

x

n  n  1 an  x  x0 

0 
2  n 
dx  n 0
 n2
9. Taylor series:

f  n   x0 
n
 x  x0  is the Taylor series for a function f(x) about the point x = x0.

n!
n 0
10. Equality of series: every corresponding term is equal…
11. Analytic functions: have a convergent Taylor series with non-zero radius of convergence
about some point x = x0.
ODE Lecture Notes
12. Shift of index of Summation:
Section 5.1
Page 6 of 6