power series

power series
an infinite series in which the terms are coefficients times successive powers of a given variable, or times products of powers of two or more variables.
[1890-95]

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      in mathematics, an infinite series that can be thought of as a polynomial with an infinite number of terms, such as 1 + x + x2 + x3 +⋯. Usually, a given power series will converge (convergence) (that is, approach a finite sum) for all values of x within a certain interval around zero—in particular, whenever the absolute value of x is less than some positive number r, known as the radius of convergence. Outside of this interval the series diverges (is infinite), while the series may converge or diverge when x = ± r. The radius of convergence can often be determined by a version of the ratio test for power series: given a general power series

a0 + a1x + a2x2 +⋯,
in which the coefficients are known, the radius of convergence is equal to the limit of the ratio of successive coefficients. Symbolically, the series will converge for all values of x such that

      For instance, the infinite series 1 + x + x2 + x3 +⋯ has a radius of convergence of 1 (all the coefficients are 1)—that is, it converges for all −1 < x < 1—and within that interval the infinite series is equal to 1/(1 − x). Applying the ratio test to the series

1 + x/1! + x2/2! + x3/3! +⋯
(in which the factorial notation n! means the product of the counting numbers from 1 to n) gives a radius of convergence of
so that the series converges for any value of x.

 Most functions can be represented by a power series in some interval (see table—>). Although a series may converge for all values of x, the convergence may be so slow for some values that using it to approximate a function will require calculating too many terms to make it useful. Instead of powers of x, sometimes a much faster convergence occurs for powers of (x − c), where c is some value near the desired value of x. Power series have also been used for calculating constants such as π and the natural logarithm base e and for solving differential equations (differential equation).
 

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Universalium. 2010.

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