When an function is analytic at c, we can find a Taylor series at point c such that the series converges to f(x) for x near c.This definition is useful because it can change a function into a infinite power series. e^x is an example for this, it is infinitely differentiable and the Taylor series is still itself.

An analytic function can be differentiable infinitely. However, a function differentiable infinitely does not mean it is analytic. Therefore, by determining if all nth derivatives are zero, we can distinguish whether an infinitely differentiable function analytic or not.