Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 11 - Infinite Sequences and Series - 11.1 Exercises - Page 726: 87

Answer

If $\lim\limits_{n\to\infty} |a_n|=0$, then $\lim\limits_{n\to\infty} a_n=0$

Work Step by Step

Here, we have the property of the absolute value: $-|a_n| \leq a_n \leq |a_n|$ for all the values of $n$ consider $\lim\limits_{n\to\infty} |a_n|=0$ Since, $-|a_n| \leq a_n \leq |a_n|$ for all the values of $n$ By the limit laws of sequences, we have $\lim\limits_{n\to\infty} |a_n|=0$; $\lim\limits_{n\to\infty} -|a_n|=-\lim\limits_{n\to\infty} |a_n|=0$ By the squeeze theorem for sequence, we have $\lim\limits_{n\to\infty} a_n=0$ Hence, If $\lim\limits_{n\to\infty} |a_n|=0$, then $\lim\limits_{n\to\infty} a_n=0$
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