University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Questions to Guide Your Review - Page 551: 14

Answer

See below.

Work Step by Step

(Theorem 9 section 3) The Integral test is applicable for the series with positive terms, such that after some index N, the terms decrease. Then, if we define a continuous real function $f$ such that $f(n)=a_{n}$, for $n\geq N$, we can test whether $\displaystyle \int_{N}^{\infty}f(x)dx$ converges or not. The series $\displaystyle \sum_{n=N}^{\infty}a_{n}$ and the integral $\displaystyle \int_{N}^{\infty}f(x)dx$ both converge or both diverge. The reasoning behind this test is that we test and determine the convergence of a series (which can be very difficult) by testing the convergence of an improper integral (which is relatively simple, or rather, better known to us). See example 4 of section 3: $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^{2}+1}$ converges because $\displaystyle \int_{1}^{\infty}\frac{1}{x^{2}+1}dx=\frac{\pi}{4}$. The sum DOES NOT equal the value $\displaystyle \frac{\pi}{4}$, but, because the integral converges, we know that the sum does as well.
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