Power series

An infinite series $\sum_{n=0}^\infty a_n(z-z_0)^n$ in powers of $(z-z_0)$ , with fixed coefficients $a_n$ .

A power series centered at $z_0$ is a series of the form $\sum_{n=0}^\infty a_n(z-z_0)^n$ , where the $a_n$ are fixed complex coefficients. For any particular $z$ this is an ordinary complex series, so it converges for some values of $z$ and diverges for others; a power series is best thought of not as one number but as a function of $z$ , defined wherever the series happens to converge.

Examples

$\sum_{n=0}^\infty z^n$ is a power series centered at $0$ ; it converges to $1/(1-z)$ for $|z|<1$ .
$\sum_{n=0}^\infty \frac{(z-i)^n}{n!}$ is a power series centered at $i$ , converging for every complex $z$ .