Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 11 - Infinite Series - 11.2 Summing and Infinite Series - Exercises - Page 547: 12

Answer

$\frac{1}{2}$

Work Step by Step

$\frac{1}{n(n-1)}$ = $\frac{1}{n-1}-\frac{1}{n}$ $\Sigma_{n=3}^{\infty}\frac{1}{n(n-1)}$ = $\Sigma_{n=3}^{\infty}(\frac{1}{n-1}-\frac{1}{n})$ $S_{N}$ = $(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})+...+(\frac{1}{N-1}-\frac{1}{N})$ = $\frac{1}{2}-\frac{1}{N}$ $S$ = $\lim\limits_{N \to \infty}S_{N}$ = $\lim\limits_{N \to \infty}(\frac{1}{2}-\frac{1}{N})$ = $\frac{1}{2}$
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