Power series

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

Related Topics:
Mathematics - Infinite series

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where the coefficients an, the center c, and the argument x are usually real or complex numbers. These series usually arise as the Taylor series of some known function; the Taylor series article contains many examples.

Related Topics:
Real - Complex - Taylor series - Function

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In many situations, the center c is equal to zero, for instance when considering a Maclaurin series.

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In such cases, the power series takes the simpler form

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::

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f(x) = sum_{n=0}^infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ldots.

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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.

Related Topics:
Generating function - Z-transform - Decimal notation - Integer - Number theory - P-adic number

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