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Integral Test for Convergence

Benjamin Wright

Benjamin Wright

5 min read

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Study Guide Overview

This guide covers the Integral Test for convergence of infinite series. It explains the theorem, including the conditions of a positive, decreasing function and its corresponding series. The guide demonstrates how to apply the integral test with examples, focusing on evaluating improper integrals and determining convergence or divergence. Practice problems and solutions are provided to reinforce the concept.

10.4 Integral Test for Convergence

Welcome to AP Calc 10.4! In this guide, you’ll learn how to apply another test to determine whether series are convergent or divergent. This test will rely heavily on your understanding of indefinite integrals from Unit 6.

∫ Integral Test Theorem

The integral test states that, for a positive, decreasing function f(x)f(x) over the interval k,∞)k,\infty) with a corresponding sequence an=f(x)a_n=f(x), then…

(1) if ∫k∞ f(x) dx\int_k^{\infty}\ f(x)\ \text{d}x converges, then ∑n=k∞an\sum_{n=k}^{\infty}a_n also converges,

(2) if ∫k∞ f(x) dx\int_k^{\infty}\ f(x)\ \text{d}x diverges, then ∑n=k∞an\sum_{n=k}^{\infty}a_n also diverges,

and…

a_k+\int_{k+1}^{\infty}\ f(x)\ \text{d}x\leq \sum_{n=k}^{\infty}a_n\leq \int_k^{\infty}\ f(x)\ \text{d}x

For now, we c...

Previous Topic - The nth Term Test for DivergenceNext Topic - Harmonic Series and p-Series

Question 1 of 9

🎉 What are the two key conditions a function f(x)f(x) must satisfy to apply the Integral Test for convergence?

Positive and increasing

Negative and decreasing

Positive and decreasing

Negative and increasing