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Taylor Series

Suppose that f is a function whose domain includes the number a and suppose that f has derivatives of all orders at a. That is, suppose that f(n)(a) exists for all n. The power series

is called the Taylor Series of f centered at a. In the case that a = 0, the Taylor Series is called the Maclaurin Series.

Example: Find the Taylor series for f(x) = 1/x2 centered at a = 1.

We have:

f(x) = 1/x2                                so         f(1) = 1
f ′(x) = –2x–3                            so         f ′ (1) = –2(1)–3 = –2

f ″(x) = 6x–4                             so         f ″(1) = 6(1)–4 = 6

f (3) (x) = –24x–5                      so         f (3) (1) = –24(1)–5 = –24

f (4) (x) = 120x–6                      so         f (4) (1) = 120(1)–6 = 120

Therefore,

 

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