# Power Series

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## power series

[′pau̇·ər ‚sir·ēz]*n*th term of the form

*a*

_{n }(

*x*-

*x*

_{0})

^{ n }, where

*x*

_{0}is some point and

*a*

_{n }some constant.

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Power Series

an infinite series of the form

*a*_{0} + *a*_{1}z +*a*_{2}z^{2} + . . . + *a*_{n}z^{n} + . . .

where the coefficients *a*_{0}, *a*_{1}, *a*_{2}, . . . , *a _{n}*, . . . are complex numbers independent of the complex variable z.

Generally speaking, the domain of convergence of a power series is an open region *D* = {*z*: ǀ*z*ǀ < *R*} bounded by a circle with center at *z* = 0. This circle is called the circle of convergence of the power series; its radius *R* is called the radius of convergence of the series. In the degenerate case where *R* = 0, the circle of convergence consists of the point *z* = 0. An example is 1 + l!*z* + 2!*z*^{2} + . . . + *n!z ^{n}* + . . . . If

*R*= ∞, the domain of convergence is the entire complex plane. An example is 1 +

*z*/1 ! +

*z*

^{2}/2! + . . . +

*z*+ . . . . The radius of convergence of a power series is expressed in terms of its coefficients in accordance with the Cauchy-Hadamard theorem:

^{n}/n!A power series converges absolutely at all points within the circle of convergence. On the circumference of the circle, where ǀ*z*ǀ = *R*, the series may either converge or diverge. For example, the series 1 + *z* + *z*^{2} + . . . + *z ^{n}* + . . . , for which

*R*= 1, diverges at each point of the circumference, where

*ǀzǀ =*1. On the other hand, the series

converges absolutely at all points of the circumference, where ǀ*z*ǀ = 1. A power series diverges at each point exterior to the circle of convergence (ǀ*z*ǀ > *R*).

Within the circle of convergence, the sum of the power series

is an analytic function. Derivatives of any order of the function /(z) can be obtained by term-by-term differentiation of the series; moreover, the power series is the Taylor series of its sum.

A. A. GONCHAR