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 10.5 - Absolute Convergence; The Ratio and Root Tests - Exercises 10.5 - Page 598: 49



Work Step by Step

Given: $a_{n+1}=(\dfrac{2}{n})a_n$ and $a_1=2$ $\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{(\dfrac{2}{n})a_n}{a_n}|$ $\implies \lim\limits_{n \to \infty}|\dfrac{2}{n}|=\dfrac{2}{\infty}=0 \lt 1$ Thus, the series converges by the ratio test.
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