Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - Chapter Review Exercises - Page 591: 31

Answer

$$ \frac{4}{37} $$

Work Step by Step

Since \begin{align*} 0.108108108 &= 0.108+0.000108+0.000000108\\ &=0.108(1)+0.12(10^{-3})+ 0.12(10^{-6})+\cdots \\ &=0.108[1+\frac{1}{10^3}+\frac{1}{10^6}+\cdots ] \end{align*} Since $$1+\frac{1}{10^3}+\frac{1}{10^6}+\cdots $$ which is a geometric series and has the sum $$\frac{1}{1-\frac{1}{1000}}= \frac{1000}{999} $$ Hence $$0.108108108= 0.108 \frac{1000}{999}= \frac{108}{999} =\frac{4}{37}$$
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