What is a Power Series?

A power series is a series that represents a function as an infinite sum of terms, each of which is a polynomial multiplied by a coefficient. The general form of a power series is:

$$sum_{n=0}^{infty} a_n cdot (x - c)^n = a_0 + a_1 cdot (x - c) + a_2 cdot (x - c)^2 + a_3 cdot (x - c)^3 + ldots$$

Key Components of a Power Series:

Terms: Each term consists of a coefficient, denoted by (a_n), multiplied by a variable raised to the (n^{th}) power.

Variable: The variable (x) represents the input to the function. The point (c) is the center of the power series and indicates where the series is expanded around.

Coefficients: The coefficients (a_n) determine the shape of the function represented by the power series. The behavior of the function is determined by how the coefficients change as (n) varies.

Applications of Power Series:

Power series are widely used in mathematics, physics, engineering, and other fields. They provide a way to represent complicated functions in a simpler form that is easier to analyze and manipulate. They are particularly useful in calculus for approximating functions, solving differential equations, and studying the behavior of functions near a specific point.