Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 11 - Infinite Sequences and Series - 11.3 Exercises - Page 745: 33

Answer

Domain : $(1,\infty)$

Work Step by Step

$\Sigma_{1}^{\infty}\frac{1}{n^{x}}=\int_{1}^{\infty}\frac{1}{t^{x}}dt$ when $x=1$ ; $\int_{1}^{\infty}\frac{1}{t^{x}}dt=\int_{1}^{\infty}\frac{1}{t}dt=lnt|_{1}^{\infty}=\infty$ when $0\lt x \lt 1$ ; $\int_{1}^{\infty}\frac{1}{t^{x}}dt=\frac{1}{1-x}[t^{(1-x)}|_{1}^{\infty}=\infty$ when $x \gt 1$ ; $\int_{1}^{\infty}\frac{1}{t^{x}}dt=\frac{1}{1-x}[t^{(1-x)}|_{1}^{\infty}=\frac{1}{1-x}(-1)=\frac{1}{x-1}$ Hence, the given function is defined when $x \gt 1$ Thus, Domain : $(1,\infty)$
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