Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 9 - Infinite Series - 9.1 Sequences - Exercises Set 9.1: 8

Answer

first five terms: $\frac{1}{3}, \frac{4}{5}, \frac{9}{7}, \frac{16}{9}, \frac{25}{11}$, diverges

Work Step by Step

To find the first five terms of the sequence, put the values $1,2,3,4,5$ into the function $a_n=\frac{n}{n+2}$. To find if the sequence converges, find the limit as n approaches positive infinity. This leaves $$\lim\limits_{n \to \infty} \frac{n^2}{2n+1}$$ Since the highest degree in the numerator is higher than the highest degree in the denominator, the limit is infinity and the sequence diverges.
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