# Taylor series

A Taylor series of a function is a special type of power series whose coefficients involve derivatives of the function. Taylor series are generally used to approximate a function, f, with a power series whose derivatives match those of f at a certain point x = c, called the center. This is because power series are relatively easy to calculate, differentiate, and integrate. With Taylor series, we can approximate values like e^{1.23} and cos(0.77), or integrate functions like , which we cannot calculate exactly.

The Taylor series for f(x) centered at c is given by:

f(x) | = | ||

= |

where f^{(n)}(c) denotes the n^{th} derivative of f at c. The 0^{th} derivative of f at c is just the value f(c). Also remember that 0! = 1, not 0, so

When c = 0, the resulting Taylor series is referred to as the Maclaurin series for f:

f(x) | = | ||

See also power series.