Differential Equations Workbook For Dummies®

8.1. Checking On a Series with the Ratio Test

Power series that become infinite aren't of much help to anyone, which is why you only work with series that stay finite in the following pages. A finite series converges to a particular value.

NOTE

A series such as the following:

is said to converge for a particular x if this limit:

is finite. It this limit is infinite, the series doesn't converge.

NOTE

How do you know whether a series converges? Just bust out the ratio test, which compares successive terms of a series to see whether the series is going to converge. If the ratio of the (n + 1)th term to the nth term is less than 1 for a fixed value of x, the series converges for that x.

For example, if you have this series:

then the ratio of the (n + 1)th term to the nth term is

The series converges if this ratio is less than 1 as n gets larger and larger.

Here's another example of the ratio test. Take a look and then check out the following problems to practice using the ratio test to determine whether a particular series converges.

NOTE

EXAMPLE

Q. Does this series converge? ...