A power series representation is a mathematical expression that represents a function as an infinite sum of terms, each of which is a power of the independent variable. It is a useful tool for approximating functions and solving differential equations.
A power series representation includes infinitely many terms, while a polynomial only has a finite number of terms. Additionally, the terms in a power series are not limited to just powers of the independent variable, but can also include other functions of the variable.
The general form of a power series representation is ∑n=0∞ an(x-c)n, where an represents the coefficient of the nth term, x is the independent variable, and c is the center of the series.
The convergence of a power series representation can be determined using the ratio test or the root test. The series will converge if the limit of the ratio or the limit of the root is less than 1. Additionally, a power series will converge within the interval of convergence, which can be determined using the ratio test.
A power series representation is useful for approximating functions and solving differential equations, as it allows for complex functions to be broken down into simpler terms. It can also be used to extend a function beyond its known domain and to study the behavior of a function near a specific point.