Power Series and Their Properties

A power series is an infinite sum that represents a function within its interval of convergence. The series is defined by coefficients multiplied by powers of x minus a, where a is the center of the series. When a equals zero, the terms are simple powers of x.

This structure makes a power series act like an infinite polynomial.

For example, setting all coefficients to one gives the geometric series with ratio x.

This series converges to 1 divided by 1 minus x only when the absolute value of x is less than 1. This defines the interval where the series behaves like the function.

Graphs of the series' partial sums, formed by adding the first few terms, closely match the function near the center.

As x moves farther from the center, adding more terms may improve accuracy. If the absolute value of x is greater than 1, the series diverges and does not represent the function.

In AC circuits, trigonometric functions model voltage and current. Power series represent these functions as infinite sums of polynomial terms. This lets circuit simulation tools calculate accurate values using basic arithmetic.