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/x^{2} centered at *a* = 1.

We have:

f(x) = 1/x^{2} 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,

_{}