Applied to a suitable function f, Taylor’s Theorem gives a polynomial that is an approximation to f(x).

Theorem

Let f be a real function on an open interval I, such that the derived functions f^{(r )}(r = 1,…, n) are continuous functions and suppose that a ε‎ I. Then, for all x in I,

$$\begin{array}{l}f(x)=f(a)+\frac{f\prime (a)}{1!}(x-a)+\frac{f″(a)}{2!}{(x-a)}^2\hfill \\ +\dots +\frac{f^{(n-1)}(a)}{(n-1)!}{(x-a)}^{n-1}+R_n,\hfill \end{array}$$

where R_n denotes the remainder term R_n.

Two possible forms for R_n are

$$\begin{array}{l}{R}_n=\frac{1}{\left(n-1\right)!}{\displaystyle {\int }_a^x{f}^{\left(n\right)}}\left(t\right){\left(x-t\right)}^{n-1}\text{d}t\text{and}\\ R_n=\frac{f^{\left(n\right)}\left(c\right)}{n!}{\left(x-a\right)}^n,\end{array}$$

where c lies between a and x. By taking x = a + h, where a + h ε‎ I, the formula

$$f\left(a+h\right)=f\left(a\right)+\frac{f\prime \left(a\right)}{1!}h+\frac{f″\left(a\right)}{2!}{h}^2+\cdots +\frac{f^{\left(n-1\right)}\left(a\right)}{\left(n-1\right)!}{h}^{n-1}+R_n$$

is obtained. This enables f(a + h) to be determined up to a certain degree of accuracy, the remainder R_n giving the error. Suppose now that f is infinitely differentiable in I and that R_n → 0 as n → ∞; then an infinite series can be obtained whose sum is f(x). In such a case, it is customary to write

$$f\left(x\right)=f\left(a\right)+\frac{f\prime \left(a\right)}{1!}\left(x-a\right)+\frac{f″\left(a\right)}{2!}{\left(x-a\right)}^2+\cdots .$$

This is the Taylor series (or expansion) for f at (or about) a. The special case with a = 0 is the Maclaurin series for f. Note that the Taylor series of an infinitely differentiable function f(x) can converge without converging to f(x); it is important that the remainder term tends to 0. For example, the function

$$f\left(x\right)=\{\begin{array}{r}\text{exp}\left(-1/x^2\right),x\ne 0,\\ 0,x=0,\end{array}$$

is infinitely differentiable at 0 with f⁽ⁿ⁾(0) = 0 for all n. Thus, the Taylor series converges, but to the zero function, rather than to f(x).

The Taylor series for a real function f(x,y) of two variables, which has partial derivatives of all orders, states that

$$\begin{array}{l}f\left(a+h,b+k\right)\\ =f\left(a,b\right)+\left(f_x\left(a,b\right)h+f_y\left(a,b\right)k\right)\\ +\frac{1}{2!}\left(f_{xx}\left(a,b\right)h^2+2f_{xy}\left(a,b\right)hk+f_{yy}\left(a,b\right)k^2\right)+\cdots .\end{array}$$

In complex analysis, a function f(z) which is holomorphic at a point a has a Taylor series

$$f\left(z\right)={\displaystyle \sum _{k=0}^{\infty }\frac{f^{\left(k\right)}\left(a\right)}{k!}{\left(z-a\right)}^k}$$

which is convergent in a neighbourhood of a. See also Cauchy’s formula for derivatives.