MTH253
Calculus III
Chapter 11, Part 2
(sections 11.7 – 11.9)
Power Series
&
Taylor Series


n 0
f
(n)
(a)
n
( x  a)
n!
Power Series about x = a

 c ( x  a)
n 0
n
n
 c0  c1 ( x  a )  c2 ( x  a ) 2 
Center: a
Coefficients: c0, c1, c2,
…
All
Constants
Essentially, a polynomial of infinite degree.
Like polynomials, power Series can be easily added,
subtracted, multiplied, differentiated, and integrated.
Example: Geometric Power Series

x
n 0
n
 1 x  x  x 
2
3
Note that this is a geometric series
with a = 1 and r = x.
When does this series converge?
What does it converge to?
That is …

1
  x n , if x  1
1  x n 0
x 1
a
1

1 r 1 x
Convergence of Power Series

For what values of x is
 c  x  a
n 0
n
n
convergent?
For any power series, one of the following will be true:
a. The series only converges when x = a.
b. The series converges absolutely for all real values of x.
c. The series converges absolutely for all –R < x-a < R and diverges
for x-a > R & x-a < -R. The behavior at x = a  R can vary.
a-R
R is called the Radius
of Convergence.
a
a+R
The values of x where the series converges
is called the Interval of Convergence.
Convergence of Power Series

For what values of x is
 c  x  a
n 0
n
n
convergent?
Basic Method:
1. Use the ratio test (limit of the absolute value of the ratio of
consecutive terms must be less than 1) and solve for x.
2. Check the endpoints using some other test and those two
specific values of x.
Example:


n 0
 3x  2 
Determine the convergence of …
n
n
Radius of Convergence?
Interval of convergence?
Operations on Power Series
If f ( x)   rn  x  a 
n
& g ( x )   sn  x  b 
n
converge absolutely for x  I , then ...
f ( x)  g ( x)
f ( x) g ( x)
d
f ( x)
dx
All converge
absolutely for x  I
 f ( x)dx
These operations can be carried out “term-by-term”.
I is the intersection of the intervals of convergence of f(x) and g(x).
Multiplying Power Series
  xn 
 
 n 0 n ! 

x 2 x3
 1  x   
2 6

 
n n
   1 x 
 n 0




1  x  x
2
 x3 

A basic goal of this chapter …
Approximate transcendental functions using
algebraic functions (specifically: polynomials).

Solution ..
• Transcendental function are differentiable of ALL
orders.
• Find a power series equal to the transcendental
function.
• Consider the interval of convergence.
• Use a portion of the series (i.e. a polynomial) to
approximate the function.
Taylor Series

p ( x)  
n 0
f
(n)
(a)
n
( x  a)
n!
f ''(a )
 f (a )  f '(a )( x  a ) 
( x  a)2
2
(4)
f (3) (a )
f
(a)
3

( x  a) 
( x  a) 4 
6
4!
NOTE: This assumes that f (n)(a) exists for all n.
Taylor Series
with a = 0
Maclaurin Series

p( x)  
n 0
f
(n)
(0) n
x
n!
f ''(0) 2 f (3) (0) 3
 f (0)  f '(0) x 
x 
x 
2
3!
NOTE: This assumes that f (n)(a) exists for all n.
Convergence of Taylor/Maclaurin Series

f ( x)  
n 0
f
( n)
(a)
n
( x  a)
n!
… for all values of x where the series converges.

Example:
f ( x)  
n 0
f ( n ) (a)
( x  a) n
n!
Find the Taylor Series & its interval of
convergence for f(x) = ln x at x = 1.
You MUST memorize these!
3 Important Maclaurin Series

k
2
3
4
x
x
x
x
ex    1  x    
2! 3! 4!
k 0 k !

x2k
x 2 x 4 x6
cos x    1
 1   
2k !
2! 4! 6!
k 0
k
x2k 1
x3 x5 x 7
sin x    1
 x   
(2k  1)!
3! 5! 7!
k 0

k
All three of these converge for all values of x.
Deriving Series for Known Series

x2k
x 2 x 4 x6
cos x    1
 1   
2k !
2! 4! 6!
k 0
cos2x 
k
Deriving Series for Known Series
x 2 x3 x 4
e  1 x    
2! 3! 4!
x
x3 x5 x 7
sin x  x    
3! 5! 7!
e sin x 
x
Deriving Series for Known Series
x 2 x3 x 4
e  1 x    
2! 3! 4!
x
x3
sin x  0  x  0 x   0 x 4 
3!
2
e sin x 
x
Other Maclaurin Series

See page 815 of the textbook.

For more, see …
http://mathworld.wolfram.com/MaclaurinSeries.html