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: 35

Answer

$$a_{n}=-1, \ \ \ \ b_{n}=\left(\frac{1}{2}\right)^{n}+1$$

Work Step by Step

Consider $$a_{n}=-1, \ \ \ \ b_{n}=\left(\frac{1}{2}\right)^{n}+1$$ Since \begin{align*} \lim_{n\to \infty } a_n &=-1\neq 0\\ \lim_{n\to \infty } b_n &= 1\neq 0 \end{align*} Then $ \sum a_n,\ \ \sum b_n $ are divergent series, but $$ \sum_{n=1}^{\infty} (a_n+b_n)=\sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^{n} $$ which is a geometric series with $r= 1/2$ and has the sum $$ \frac{a}{1-r}= \frac{\frac{1}{2}}{1-\frac{1}{2}}=1$$
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