A power series is an infinite polynomial-like expression centered at a point $c$:

$$\sum_{n=0}^{\infty}a_n(x-c)^n.$$

For each fixed value of $x$, this becomes an ordinary numerical series. Whether it converges therefore depends on how far $x$ is from the center.

Radius of convergence

A power series has a radius of convergence $R$ with one of the following possibilities:

• it converges only at $x=c$ when $R=0$;
• it converges for every real $x$ when $R=\infty$;
• or, for a finite positive $R$, it converges whenever

$$|x-c|<R$$

and diverges whenever $|x-c|>R$.

The ratio or root test often reveals $R$.

Endpoints

When $R$ is finite, the points

$$x=c-R\qquad\text{and}\qquad x=c+R$$

must be checked separately. One endpoint may converge while the other diverges, so the interval of convergence contains more information than the radius alone.

A power series as a function

Inside its interval of convergence, a power series defines a function:

$$f(x)=\sum_{n=0}^{\infty}a_n(x-c)^n.$$

Within the interior of that interval, power series can be differentiated and integrated term by term while retaining the same radius of convergence.

This makes them a bridge between infinite series and the local representation of functions.