Taylor Series

# To see why this is the case note that if we x an x0

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: r two inequalities together to get a rather messy inequality: −( x4 x4 M1 M1 x4 + M2 )x8 ≤ (e 2 − (1 + ) − (cos(x2 ) − (1 − )) ≤ ( + M2 )x8 4 2 2 4 However, if we let M = M1 4 + M2 and simplify, this inequality becomes x4 −M x8 ≤ e 2 − cos(x2 ) − x4 ≤ M x8 for all x ∈ [−1, 1]. Dividing by x4 gives us that x4 e 2 − cos(x2 ) − 1 ≤ M x4 −M x ≤ x4 4 The ﬁnal step is to use the Squeeze Theorem to show that x4 e 2 − cos(x2 ) −1=0 lim x→0 x4 or equivalently that x4 e 2 − cos(x2 ) = 1. lim x→0 x4 This limit is conﬁrmed by the graph of the function h(x) = DE Math 128 334 e x4 2 −cos(x2 ) . x4 (B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series This is a rather complicated looking argument. However, with a little practice using Taylor polynomials and the mastery of a few tricks, limits like this can actually be done by inspection! 4.5.3 Convergence of Taylor Series In this section we will return to a question that we asked earlier: Question: Given a functionf (x) that is inﬁnitely diﬀerentiable at x = a, is f (x) equal t...
View Full Document

## This document was uploaded on 03/23/2013.

Ask a homework question - tutors are online