Power Series

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

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

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 ⅟10. In number theory, the concept of p-adic numbers is also closely related to that of a power series.

Read more about Power Series:  Examples, Radius of Convergence, Analytic Functions, Formal Power Series, Power Series in Several Variables, Order of A Power Series

Other articles related to "power series, series":

Logrithm - Calculation - Power Series
... Taylor series For any real number z that satisfies 0 < z < 2, the following formula holds This is a shorthand for saying that ln(z) can be approximated to a more and more ... This series approximates ln(z) with arbitrary precision, provided the number of summands is large enough ... In elementary calculus, ln(z) is therefore the limit of this series ...
Struve Function - Definitions - Power Series Expansion
... Struve functions, denoted as have the following power series form where is the gamma function ... The modified Struve function, denoted as have the following power series form ...
Order of A Power Series
... Let α be a multi-index for a power series f(x1, x2, …, xn) ... The order of the power series f is defined to be the least value

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