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: 5

Answer

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.
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