Editing Series (mathematics) (section)

===Power series===
:{{Main|Power series}}

A '''power series''' is a series of the form

<math display=block>\sum_{n=0}^\infty a_n(x-c)^n.</math>

The [[Taylor series]] at a point {{tmath|c}} of a function is a power series that, in many cases, converges to the function in a neighborhood of {{tmath|c}}. For example, the series

<math display=block>\sum_{n=0}^{\infty} \frac{x^n}{n!}</math>

is the Taylor series of <math>e^x</math> at the origin and converges to it for every {{tmath|x}}.

Unless it converges only at {{tmath|1= x = c}}, such a series converges on a certain open disc of convergence centered at the point {{tmath|c}} in the complex plane, and may also converge at some of the points of the boundary of the disc.  The radius of this disc is known as the [[radius of convergence]], and can in principle be determined from the asymptotics of the coefficients {{tmath|a_n}}.  The convergence is uniform on [[closed set|closed]] and [[bounded set|bounded]] (that is, [[compact set|compact]]) subsets of the interior of the disc of convergence: to wit, it is [[Compact convergence|uniformly convergent on compact sets]].

Historically, mathematicians such as [[Leonhard Euler]] operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.