RaySix

Post

Understanding Power Series and Convergence

Created by @adamvaughn

at November 5th 2023, 7:55:20 pm.

Calculus

#convergence

#power-series

#radius-of-convergence

Understanding Power Series

A power series is a representation of a function as an infinite sum of terms that are a power of a variable. It is a powerful tool in calculus that allows us to represent a wide range of functions in a concise and efficient manner. In this post, we will explore the concept of power series, discuss their convergence and divergence, and introduce the concept of the radius of convergence.

Definition

A power series centered at a point c is given by the following formula:

f(x) = \sum_{n=0}^{\infty} a_n (x-c)^n

where a_n are constant coefficients and x is the variable.

The terms in the power series represent different powers of (x - c). By adjusting the coefficients a_n, we can approximate a function f(x) using a specific number of terms in the series.

Convergence and Divergence

The convergence of a power series can vary depending on the values of x and c. A power series may converge for some values of x and diverge for others.

The power series will converge if the following limit exists:

L = \lim_{n\to\infty} | \frac{a_{n+1}}{a_n}|

If L is less than 1, the power series will converge. If L is greater than 1 or the limit does not exist, the power series will diverge.

Radius of Convergence

The radius of convergence, denoted by R, is a crucial concept when dealing with power series. It represents the interval around the center point c within which the power series converges.

The radius of convergence can be determined using the following formula:

R = \frac{1}{\lim_{n\to\infty} |\frac{a_{n+1}}{a_n}|}

Example

Consider the function f(x) = \frac{1}{1-x}. We can represent this function as a power series centered at c = 0:

f(x) = \sum_{n=0}^{\infty} x^n

In this case, the coefficients a_n are all equal to 1. We can see that this power series is geometric, and it converges when |x| < 1.

Let's approximate the function f(x) by using the first four terms of the power series:

f(x) = 1 + x + x^2 + x^3

By substituting various values of x within the interval |x| < 1, we can obtain increasingly accurate approximations of f(x).

Power series provide a valuable tool for approximating functions and performing calculations. In the next post, we will dive deeper into the Taylor series expansion, a special case of the power series, and its applications in calculus.