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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
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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