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

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

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