Power Series

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power series

[′pau̇·ər ‚sir·ēz]
An infinite series composed of functions having n th term of the form an (x-x0) n , where x0 is some point and an some constant.

Power Series


an infinite series of the form

a0 + a1z +a2z2 + . . . + anzn + . . .

where the coefficients a0, a1, a2, . . . , an, . . . 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: ǀzǀ < 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!z2 + . . . + n!zn + . . . . If R = ∞, the domain of convergence is the entire complex plane. An example is 1 + z/1 ! + z2/2! + . . . + zn/n! + . . . . The radius of convergence of a power series is expressed in terms of its coefficients in accordance with the Cauchy-Hadamard theorem:

A power series converges absolutely at all points within the circle of convergence. On the circumference of the circle, where ǀzǀ = R, the series may either converge or diverge. For example, the series 1 + z + z2 + . . . + zn + . . . , 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 ǀzǀ = 1. A power series diverges at each point exterior to the circle of convergence (ǀzǀ > 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.


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