Power series

$\sum c_n(x-a)^n$ — an infinite series that's also a function of $x$ , generalizing a polynomial to infinitely many terms.

A power series centered at $a$ is $\sum_{n=0}^{\infty}c_n(x-a)^n$ — a series whose terms involve increasing powers of $(x-a)$ , making it a FUNCTION of $x$ rather than a fixed number. For each value of $x$ plugged in, the power series becomes an ordinary numerical series, which may converge or diverge depending on $x$ .

Examples

$\sum_{n=0}^{\infty}x^n$ is a power series centered at $a=0$ ; plugging in $x=\frac12$ gives the numerical geometric series $\sum\left(\frac12\right)^n$ .