Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.1 Sequences - 11.1 Exercises - Page 746: 87


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

Work Step by Step

Prove that If $\lim\limits_{n\to\infty} |a_n|=0$, then $\lim\limits_{n\to\infty} a_n=0$ Consider $\lim\limits_{n\to\infty} |a_n|=0$ The property of absolute value is defined as: $-|a_n| \leq a_n \leq |a_n|$ for all the values of $n$. Apply Law of sequences for the limits. $\lim\limits_{n\to\infty} |a_n|=0$; $\lim\limits_{n\to\infty} -|a_n|=-\lim\limits_{n\to\infty} |a_n|=0$ According to the squeeze theorem for sequence, we have $\lim\limits_{n\to\infty} a_n=0$ Hence, it has been prove that when $\lim\limits_{n\to\infty} |a_n|=0$, then $\lim\limits_{n\to\infty} a_n=0$
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