Power series

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

where an 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

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 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.

Contents

Examples

Any polynomial can be easily expressed as a power series around any center c, albeit one with most coefficients equal to zero. For instance, the polynomial f(x) = x2 + 2x + 3 can be written as a power series around the center c = 0 as

or around the center c = 1 as

or indeed 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 | x | < 1, is one of the most important examples of a power series, as are the exponential function formula

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