Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.4 The Comparison Tests - 11.4 Exercises - Page 771: 35


$error \leq 6.4 \times 10^{-8}$ and first ten terms $\approx 0.0739293$

Work Step by Step

$\Sigma_{n=1}^{\infty} 5^{-n}cos^{2}n=\Sigma_{n=1}^{\infty}\frac{cos^{2}n}{5^{n}}$ $T_{n}=\int_{10}^{\infty}\frac{1}{5^{x}}dx=\lim\limits_{t \to \infty} \int_{10}^{t}\frac{1}{5^{x}}dx$ $=\lim\limits_{t \to \infty} [-\frac{5^{-x}}{ln5}]_{10}^{t}$ $\approx 6.4 \times 10^{-8}$ Using a calculator or by hand we can sum the first 10 terms: $\Sigma_{n=1}^{\infty}\frac{cos^{2}n}{5^{n}}\approx 0.0739293$ Hence, $error \leq 6.4 \times 10^{-8}$ and first ten terms $\approx 0.0739293$
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