Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.1 Sequences - Preliminary Questions - Page 537: 5


a) False b) Follow from Theorem 5 c) False

Work Step by Step

a) This statement is false. The sequence $a_n=cos(\pi n)$ is bounded since $-1 < cos \pi n \leq 1$ for all $n$, but it does not converge: since $a_n=cos(\pi n)=(-1)^n$, the terms assume the two values 1 and -1 alternatively, hence they do not approach one value. b) By Theorem 5, a converging sequence must be bounded. Therefore, if a sequence is not bounded, it certainly does not converge. c) The statement is false. The sequence $a_n=(-1)^n$ is bounded, but it does not approach one limit.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.