If this series converges for every *x* in the interval (*a*-*r*, *a*+*r*) and the sum is equal to *f*(*x*), then the function *f*(*x*) is called **analytic**. To check whether the series converges towards *f*(*x*), one normally uses estimates for the remainder term of Taylor's theorem. A function is analytic if and only if it can be represented as a power series; the coefficients in that power series are then necessarily the ones given in the above Taylor series formula.

If *a* = 0, the series is also called a **Maclaurin series**.

The importance of such a power series representation is threefold. First, differentiation and integration of power series can be performed term by term and is hence particularly easy. Second, an analytic function can be uniquely extended to a holomorphic function defined on an open disk in the complex plane, which makes the whole machinery of complex analysis available. Third, the (truncated) series can be used to compute function values approximately.

Some functions cannot be written as Taylor series because they have a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable *x*; see Laurent series. For example, *f*(*x*) = exp(−1/*x*²) can be written as a Laurent series.

The Parker-Sockacki theorem is a recent advance in finding Taylor series which are solutions to differential equations. This theorem is an expansion on the Picard iteration.

Exponential function and natural logarithm:

The Taylor series may be generalised to functions of more than one variable with