Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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) ba 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 ( n1) (c) Rn ( x) ( x a)n1 (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 ( n1) (c) Rn ( x) ( x a)n1 (n 1)! t between x and a , then Rn ( x) M xa n 1 (n 1)! Application Example Rn ( x) M xa 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 n1 M n1 [cos x] dx then Rn ( x) 1 M 1 n 1 [ 2 ]n1 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)).