Transcript Convergence of Taylor Series

MTH 253
Calculus (Other Topics)
Chapter 11 – Infinite Sequences and Series
Section 11.9 – Convergence of Taylor Series;
Error Estimates
Copyright © 2009 by Ron Wallace, all rights reserved.
The Mean Value Theorem
f (b)  f (a )
 f '(c)
ba
OR …
Letting b = x …
From section 4.2
for some c between a and b
f (b)  f (a)  f '(c)(b  a)
f ( x)  f (a)  f '(c)( x  a )
for some c between a and x
Taylor’s Formula
A generalization of the
Mean Value Theorem
f ( x)  Pn ( x)  Rn ( x)
where …
Remainder
n
Pn ( x)  
k 0
f ( k ) (a)
( x  a) k
k!
of order
n.
f ( n1) (c)
Rn ( x) 
( x  a)n1
(n  1)!
for some c between a and x

If lim Rn ( x)  0 x  I , then f ( x)  
n
k 0
f ( k ) ( a)
( x  a) k
k!
Remainder Estimate
Remainder
of order
n.
If M  f ( n 1) (t )
f ( n1) (c)
Rn ( x) 
( x  a)n1
(n  1)!
 t between x and a ,
then
Rn ( x)  M
xa
n 1
(n  1)!
Application Example
Rn ( x)  M
xa
n 1
(n  1)!
How many terms of its Maclaurin series are
needed to approximate cos x with an error less
than 0.001?
d n 1
M  n 1 [cos x]
dx
then

Rn ( x)  1
M 1
n 1
[ 2 ]n1

 0.001
(n  1)! (n  1)!
x
n = 6 gives 0.00468
n = 7 gives 0.00092
Generating Taylor Series
 Given known Taylor Series, other
series can be obtained using the
following operations term by term …
 Substitution
 Addition & Subtraction
 Multiplication
 Differentiation
 Integration
Generating Taylor Series
Example 1
1
 1  x  x 2  x3 
1 x
let x = -x2
1
2
4
6

1

x

x

x

2
1 x
integrate
x3 x5 x 7
tan x  x    
3 5 7
1
Much easier than finding a general formula for
the nth derivative of the tan-1x function.
Generating Taylor Series
Example 2
2
3
4
x
x
x
ex  1  x   
2 6 24
x2 x4
cos x  1  

2 24
Multiply
x3 x 4
e cos x  1  x   
3 6
x
Much easier than finding derivatives excosx … try it?
Some important Taylor & Maclaurin Series
See the list on page 815.
Some things to note:
• Don’t forget to consider the interval of convergence.
• Some converge quickly (esp. w/ n! involved).
• Some converge slowly (e.g. ln(1+x)).