Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 10: Infinite Sequences and Series - Questions to Guide Your Review - Page 635: 10

Answer

See below.

Work Step by Step

Theorem 8 deals with these types of properties for $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}$ series: If $\displaystyle \sum_{n=1}^{\infty}a_{n}=A$ and $\displaystyle \sum_{n=1}^{\infty}b_{n}=B$ are convergent series, then 1. Sum Rule: $\displaystyle \sum_{n=1}^{\infty}(a_{n}+b_{n})=\sum_{n=1}^{\infty}a_{n}+\sum_{n=1}^{\infty}b_{n}=A+B.$ 2. Difference Rule: $\displaystyle \sum_{n=1}^{\infty}(a_{n}-b_{n})=\sum_{n=1}^{\infty}a_{n}-\sum_{n=1}^{\infty}b_{n}=A-B.$ 3. Constant Multiple Rule: $\displaystyle \sum_{n=1}^{\infty}ka_{n}=k\sum_{n=1}^{\infty}a_{n}=kA$ for any number $k.$ And, if the series in question are $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}$, we have the Corollary to theorem 8: 1. Every nonzero constant multiple of a divergent series diverges. 2. If $\displaystyle \sum_{=1}^{\infty}a_{n}$ converges and $\displaystyle \sum_{n=1}^{\infty}b_{n}$ diverges, then $\displaystyle \sum_{n=1}^{\infty}(a_{n}+b_{n})$ and $\displaystyle \sum_{n=1}^{\infty}(a_{n}-b_{n})$ both diverge.
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