N 310 b forrest2 chapter 4 sequences and series 45

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Unformatted text preview: f (n) (a) (x0 − a)n ? n! 310 (B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series We know the answer to the first problem since we have developed a method for finding the interval of convergence of a power series. The second problem seems intuitively to be true. However, a closer look at what this says reveals why it may not be so. Essentially what we are asking is to be able to rebuild exactly a function over an interval that could very well be the whole real line by using only the information provided by the function at one single point. In this respect, it would seem that the fact that we can use information about ex at x = 0 to get that ∞ x e= n=0 xn n! for all x seems quite remarkable and indeed it is! To further illustrate why ex is such a special function, consider 1 e if x < −1 ex if − 1 ≤ x ≤ 1 g (x) = e if x > 1 On the interval [−1, 1], g (x) behaves exactly like ex . In particular, g (0) = e0 = 1 and g (n) (0) = e0 = 1 for every n. This means that the Taylor series ce...
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This document was uploaded on 03/23/2013.

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