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# Power series

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

- [math]f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c) + a_2 (x-c)^2 + a_3 (x-c)^3 + \cdots[/math]

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 arises as the Taylor series of some known function; the Taylor series article contains many examples.

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

- [math] f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots. [/math]

These power series arise primarily in analysis, but also occur 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.

[[Category:Mathematics