The ratio test

 Objectives:

1) Define the ratio test

2) Use the ratio test to determine if an infinite series is convergent or divergent

Definition 

The ratio test uses the limit of the absolute value of the ratio of consecutive terms of an infinite series to determine if a series is convergent  or divergent or inconclusive.

Theorem

Examples

For each of the following series, use the ratio test to determine if the series is convergent or divergent

Solution

Example a.

Let's calculate ρ:

Example b

Practice

Use the ratio test to determine if the following series is convergent or divergent.