Section 10.10

The Ratio Test

A powerful tool for series involving factorials and exponentials.

1

Introduction

Some series, like those with or , are hard to integrate or compare. The Ratio Test compares each term to the next one to see if the series behaves like a geometric series.

Interactive: Convergence of Ratios

Green: Ratio for approaches 0 (< 1, Converges).
Red: Ratio for grows unbounded (> 1, Diverges).
The black line is (The cutoff).

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The Ratio Test

Definition

Let be a series. Define:
  • If : The series is Absolutely Convergent.
  • If (or ): The series is Divergent.
  • If : The test is Inconclusive (could be anything).

Note on L=1: This typically happens for rational functions (fractions with polynomials) like or . The Ratio Test is usually useless for these; use Comparison or Integral tests instead.

3

Worked Examples

Example 1: Factorials

Determine convergence of . (Wait, let's do a factorial one as promised). Let's do .

Calculate ratio :

Recall .

Limit as :
.

Since , the series Converses Absolutely.

Example 2: Exponentials

Test .

Ratio:

Group terms:

(as n goes to infinity)
.

Since , the series Diverges.

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Practice Quiz

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