Power series

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

$f(x)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{(x-c)}^n=a_0+a_1(x-c)+a_2(x-c{)}^2+a_3(x-c{)}^3+\cdots$

where a_n represents the coefficient of the nth term, c is a constant, and x varies around c (for this reason one sometimes speaks of the series as being centered at c). This series usually appears as the Taylor series of some known function; the Taylor series article contains many examples.

In many situations c is equal to zero, for example when considering a Maclaurin series. In those cases, the power series takes the simpler form

$f(x)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots .$

These power series appear primarily in analysis, but also appear in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the Z-transform). The familiar decimal notation for integers can also be viewed as an example of a power series, but with the argument x fixed at 10. In number theory, the concept of p-adic numbers is also closely related to that of a power series.