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 e1.23 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:
where f(n)(c) denotes the nth derivative of f at c. The 0th 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:
See also power series.