AP Calculus BC Study Guide & Notes

Introduction

Welcome to this guide on Ratio Test for Convergence, this article will guide you through how to apply the ratio test to test for convergence.

Consider an infinite series $a_{n}$ , if

If $n \to \infty lim \frac{a _{n + 1}}{a _{n}} < 1$ , then the series converges.
If $n \to \infty lim \frac{a _{n + 1}}{a _{n}} > 1$ , the series diverges.
If $n \to \infty lim \frac{a _{n + 1}}{a _{n}} = 1$ , this test cannot draw any conclusion about the convergence and divergence of the series.

Example 1:

Determine the convergence of the series:

$n = 1 \sum \infty \frac{2 ^{n}}{n !}$

Apply the ratio test:

$n \to \infty lim \frac{2 ^{n + 1}}{( n + 1 )!} \frac{n !}{2 ^{n}}$

$= n \to \infty lim \frac{2}{( n + 1 )}$

$= 0 < 1$

Meaning this series converges.

Example 2:

Determine the convergence of this series:

$n = 1 \sum \infty \frac{n ^{2} 2 ^{n + 1}}{3 ^{n}}$

Apply the ratio test:

$n \to \infty lim \frac{( n + 1 ) ^{2} 2 ^{n + 2}}{3 ^{n + 1}} \frac{3 ^{n}}{n ^{2} 2 ^{n + 1}}$

$= n \to \infty lim \frac{2}{3} (\frac{n + 1}{n})^{2}$

$= \frac{2}{3} (n \to \infty lim \frac{n + 1}{n})^{2}$

$= \frac{2}{3} < 1$

Meaning this series converges.

Determine if the following series converge or diverge using the ratio test: