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Concept# Power series

Summary

In mathematics, a power series (in one variable) is an infinite series of the form
where an represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth 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
Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic 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.
Any polynomial can be easily expressed as a power series around any center c, although all but finitely many of the coefficients will be zero since a power series has infinitely many terms by definition. For instance, the polynomial can be written as a power series around the center as
or around the center as
This is because of the Taylor series expansion of f(x) around is
as and the non-zero derivatives are , so and , a constant.
Or indeed the expansion is possible around any other center c. One can view power series as being like "polynomials of infinite degree," although power series are not polynomials.
The geometric series formula
which is valid for , is one of the most important examples of a power series, as are the exponential function formula
and the sine formula
valid for all real x.
These power series are also examples of Taylor series.
Negative powers are not permitted in a power series; for instance, is not considered a power series (although it is a Laurent series).

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