Power Series

In mathematics, a power series is an infinite series of the form:

Σ a_n (x - c)^n = a_0 + a_1(x - c) + a_2(x - c)^2 + a_3(x - c)^3 + ...

for n = 0 to infinity, where x is a variable, c is a constant known as the center of the series, and a_n represents the coefficients of the series.

A power series in one variable x about 0 (i.e., c = 0) is often called a Maclaurin series, while a power series about c is often called a Taylor series.

Convergence of Power Series

A power series will converge (i.e., have a sum) for some values of x and may diverge for others. The set of all x for which the series converges is known as the interval of convergence. The radius of convergence R is the radius of the interval centered at c that contains all x for which the series converges.

Applications of Power Series

Power series are used throughout mathematics, physics, and engineering because they can be used to represent a wide range of functions, and because they often simplify calculations.

In calculus, power series are used to define and compute complex functions, such as exponential, logarithmic, and trigonometric functions. They can also be used to solve differential equations and to approximate functions.

In physics, they are used in the study of wave behavior and quantum mechanics, among other applications.