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,