Topic: Sequences and Infinite Series

Subtopic: Ratio Test & Root Test

**Overview**

The ratio test and root test are two of the most common tests to use to determine if a series is convergent or divergent. The ratio test involves taking the limit of a ratio and the root test involves taking the limit of a root. The ratio test is advisable when the n-th term of the series involves n!, a^n, nor n^n where a is a constant. The root test is advisable when the n-th term of the series involves a power containing n.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 11.6.1 Understand the conditions necessary to apply the ratio test and know the types of series for which the ratio test is a good choice
- 11.6.2 Know how to use the ratio test accurately to determine the convergence or divergence of a series
- 11.6.3 Understand when the ratio test is inconclusive
- 11.6.4 Understand the conditions necessary to apply the root test and know the types of series for which the root test is a good choice
- 11.6.5 Know how to use the root test accurately to determine the convergence or divergence of a series
- 11.6.6 Understand when the root test is inconclusive

**Terminology**

Define: ratio test, root test, limit "L" from the ratio or root test, inconclusive

**Supplemental Resources (optional)**

Paul's OL Notes - Calc II: Ratio Test and Root Test

Selwyn Hollis's Video Calculus:

Comparison Tests (including the ratio and root tests)

Patrick JMT Just Math Tutorials:

Ratio Test - Example 1

Ratio Test - Example 2

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down middle column to "Infinite Series" for related titles.