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Comparison Tests for Convergence

Hannah Hill

Hannah Hill

6 min read

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Study Guide Overview

This guide covers comparison tests for convergence of series in AP Calculus BC. It explains the Direct Comparison Test and the Limit Comparison Test, including conditions for their use. It provides practice examples demonstrating how to apply these tests, using techniques like the p-series test, geometric series test, and L’Hopital’s Rule.

10.6 Comparison Tests for Convergence

Welcome to AP Calc 10.6! In this lesson, you’ll how to test for convergence by comparing your function to an easier one!

➕ Comparison Test Theorems

In calculus, we use comparison tests when we are dealing with a series that is too complicated to determine the convergence of directly. There are two types of comparison tests, so we’ll break them both down here!

➡️ Direct Comparison Test

The Direct Comparison Test states that for two series, ∑an\sum a_n and ∑bn\sum b_n where an, bn≥0a_n,\ b_n\geq0 and an≤bna_n\leq b_n,

(1) ∑an\sum a_n converges if ∑bn\sum b_n converges and

(2) ∑bn\sum b_n diverges if ∑an\sum a_n.

Let’s think this through for a second and put it in plain English! We have two series, both of which have to be positive (an, bn≥0a_n,\ b_n\geq0 is our first condition). Our first function must be smaller than our second function (an≤bna_n\leq b_n is our second condition). If our larger series converges, then a smaller one must also converge. Likewise, if our smaller series diverges, a larger series must also diverge.

We’ll get into when this test is appropriate to use in the examples!

🔁 Limit Comparison Test

Sometimes we can’t directly compare a function ...

Question 1 of 8

🚀 Which of the following series would be most suitable to directly compare with ∑n=1∞1n2+3\sum_{n=1}^{\infty} \frac{1}{n^2 + 3} using the Direct Comparison Test?

∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n}

∑n=1∞1n3\sum_{n=1}^{\infty} \frac{1}{n^3}

∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}

∑n=1∞12n\sum_{n=1}^{\infty} \frac{1}{2^n}