Editing Power series (section)

{{other uses}}
{{short description|Infinite sum of monomials}}

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In [[mathematics]], a '''power series''' (in one [[variable (mathematics)|variable]]) is an [[infinite series]] of the form
<math display="block">\sum_{n=0}^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots</math>
where ''<math>a_n</math>'' represents the [[coefficient]] of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in [[mathematical analysis]], where they arise as [[Taylor series]] of [[infinitely differentiable function]]s. In fact, [[Borel's lemma|Borel's theorem]] implies that every power series is the Taylor series of some smooth function.

In many situations, the center ''c'' is equal to zero, for instance for [[Maclaurin series]]. In such cases, the power series takes the simpler form
<math display="block">\sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots.</math>

The [[partial sum]]s of a power series are [[polynomial]]s, the partial sums of the Taylor series of an [[analytic function]] are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power series with only finitely many non-zero terms.

Beyond their role in mathematical analysis, power series also occur in [[combinatorics]] as [[generating function]]s (a kind of [[formal power series]]) and in electronic engineering (under the name of the [[Z-transform]]). The familiar [[Decimal representation|decimal notation]] for [[real number]]s can also be viewed as an example of a power series, with [[integer]] coefficients, but with the argument ''x'' fixed at {{Fraction|1|10}}. In [[number theory]], the concept of [[p-adic number|''p''-adic numbers]] is also closely related to that of a power series.