Power series

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

where a_n 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

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  ¹⁄₁₀. In number theory, the concept of p-adic numbers is also closely related to that of a power series.

Any polynomial can be easily expressed as a power series around any center c, although most of the coefficients will be zero since a power series has infinitely many terms by definition. For instance, the polynomial ${\displaystyle f(x)=x^{2}+2x+3}$ can be written as a power series around the center ${\displaystyle c=0}$ as

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