Power series

A power series is a series of the form $\sum_{n=0}^\infty c_n (x-a)^n$ with real coefficients $c_n$ and a real center $a$.

There exists a number $R\in[0,\infty]$, called the radius of convergence, such that the series converges for $|x-a|<R$ and diverges for $|x-a|>R$ (the endpoint behavior at $|x-a|=R$ is decided separately). The radius can often be computed using the Cauchy–Hadamard theorem .

Examples:

• The geometric power series $\sum_{n=0}^\infty x^n$ converges for $|x|<1$ and represents the function $\frac{1}{1-x}$ on that interval.
• The exponential series $\sum_{n=0}^\infty \frac{x^n}{n!}$ converges for every real $x$.