Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 11 - Section 11.2 - Series - 11.2 Exercises - Page 718: 87

Answer

$\Sigma a_{n}$ is convergent.

Work Step by Step

Suppose that a series $\Sigma a_{n}$ has positive terms. So $s_{n} - s_{n-1} = a_{n} \gt 0$ for all $n$ $\to$ $s_{n} \gt s_{n-1}$ for all $n$ Thus {$s_{n}$} is an increasing sequence. Since $s_{n} \leq 1000$ for all of $n$ So {$s_{n}$} is bounded sequence. We know that every monotonic and bounded sequence is convergent. So {$s_{n}$} converges, and the sequence of partial sums is convergent. And then the series $\Sigma a_{n}$ is convergent.
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