Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 10: Infinite Sequences and Series - Section 10.1 - Sequences - Exercises 10.1 - Page 570: 57


converges to $1$

Work Step by Step

As we know that when $x \gt 0$ so, $\lim\limits_{n \to \infty} \sqrt[n] {n}=1$ and $\lim\limits_{n \to \infty} x^{1/n}=1$ Let $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} (\dfrac{3}{n})^{1/n}$ This implies that $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} (\dfrac{3}{n})^{1/n}$ and $a_n=\dfrac{\lim\limits_{n \to \infty} 3^{(1/n)}}{\lim\limits_{n \to \infty} n^{(1/n)}}=\dfrac{1}{1}=1$ Thus, $\lim\limits_{n \to \infty} a_n=1$ and {$a_n$} converges to $1$
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