Power Series

power series

[′pau̇·ər ‚sir·ēz]

(mathematics)

An infinite series composed of functions having n th term of the form a_n (x-x₀) ⁿ , where x₀ is some point and a_n some constant.

Power Series

 

an infinite series of the form

a₀ + a₁z +a₂z² + . . . + a_nzⁿ + . . .

where the coefficients a₀, a₁, a₂, . . . , 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: 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.

A. A. GONCHAR