Integral Test for Convergence

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 over the interval with a corresponding sequence , then…
(1) if converges, then also converges,
(2) if diverges, then 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...

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