Maclaurin and Taylor series

For a (real) function f under certain conditions (Taylor's Theorem)

f(x) = f(a) + (x - a)f'(a) + (x - a)²f⁽²⁾(a)/2! + ... + (x - a)ⁿf⁽ⁿ⁾(a)/n! + R_n

One obtains a Maclaurin series when a = 0. However, introducing g(x) = f(x + a) one gets f⁽ⁿ⁾(a) = g⁽ⁿ⁾(0), and so the Maclaurin series for g at x = 0 coincides with the Taylor series of f at x = a.

The remainder R_n looks very much like the expected next term, with the derivative evaluated at an intermediate point:

R_n = (x - a)^{n + 1}f^{(n + 1)}(γ)/(n + 1)!,

where γ is a point between a and x. For the derivation of this form for the remainder of the series f is required to have at least n + 1 continuous derivatives.

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