A power series (centered at the origin) is an expression of the form: $\displaystyle \sum_{n=0}^{\infty} a_n x^n = a_0+a_1x+a_2x^2+a_3x^3+...$ where $x$ is our variable and each $a_i$ is a fixed number. Examples include $\displaystyle \sum_{n=0}^\infty x^n/n!$ and $\sum_{n=0}^\infty x^n/(n+1)$.

The set of $x$ where the series converges is called the interval of convergence, and is an interval from $-R$ to $R$, where $R$ is a called the radius of convergence. The interval sometimes includes one, both, or no endpoints. You have to check separately.

Special cases:

• If $R=0$, then the series converges at the single point $x=0$;
• If $R=\infty$, then the series converges on the entire real line.