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The nth Term Test for Divergence

Benjamin Wright

Benjamin Wright

4 min read

Next Topic - Integral Test for Convergence

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Study Guide Overview

This study guide covers the nth Term Test for Divergence for AP Calculus BC. It explains how to apply the test to determine if a series diverges by evaluating the limit of the nth term as n approaches infinity. It includes a walkthrough example and practice problems with solutions, focusing on converting to limit notation, evaluating the limit, and drawing conclusions based on the test results.

#10.3 The nth Term Test for Divergence

Welcome back to Unit 10 of AP Calculus BC! Today, we’re going to discuss the nth-term test for divergence with series. Let’s get started!

#🤷‍♀️ What is the nth Term Test for Divergence?

As the name suggests the nth Divergence test tells us if a series will diverge! (mind-blowing stuff guys, I know 🤯). The Divergence test states that:

if lim⁡n→∞an≠0,∑an diverges\text{if} \ \displaystyle{\lim_{n \to \infty}} a_n \not = 0, \sum a_n \ \text{diverges}if n→∞lim​an​=0,∑an​ diverges

As we can see, if the nth term doesn't approach 0, the series diverges. On the other...

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Previous Topic - Working with Geometric SeriesNext Topic - Integral Test for Convergence

Question 1 of 8

Which of the following correctly states the nth term test for divergence? 🤔

If lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0limn→∞​an​=0, then the series ∑an\sum a_n∑an​ diverges

If lim⁡n→∞an≠0\lim_{n \to \infty} a_n \neq 0limn→∞​an​=0, then the series ∑an\sum a_n∑an​ converges

If lim⁡n→∞an≠0\lim_{n \to \infty} a_n \neq 0limn→∞​an​=0, then the series ∑an\sum a_n∑an​ diverges

If lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0limn→∞​an​=0, then the series ∑an\sum a_n∑an​ converges