What is "power series"

n. (context mathematics English) An infinite series in form $sum\_0^infty\; a\_i\; (x-c)^n$.

n. the sum of terms containing successively higher integral powers of a variable

In mathematics, a power series (in one variable) is an infinite series of the form

∑a(x − c) = a + a(x − c) + a(x − c) + …
where a represents the coefficient of the nth term and c is a constant. This series usually arises as the Taylor series of some known function.

In many situations c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form

∑ax = a + ax + ax + ⋯.
These power series arise primarily in analysis, but also occur in combinatorics (as generating functions, a kind of formal power series) and in electrical engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at . In number theory, the concept of p-adic numbers is also closely related to that of a power series.