Chapter 8: Infinite Sequences and Series

Section 8.3: Convergence Tests

Example 8.3.4

$$ Determine if the series $\sum _{n\=1}^{\infty }\frac{n}{\sqrt{n^2\+2}}$ diverges, converges absolutely, or converges conditionally. If it converges conditionally, determine if it also converge absolutely. $$ Solution $$ For large $n$, $n\/\sqrt{n^2\+2}$ behaves like $n\/n\=1$, suggesting that the limiting behavior of $a_n$ be tested by   $\underset{n\to \infty }{lim}\frac{n}{\sqrt{n^2\+2}}$ = $1$    By the nth-term test, since $a_n$ does not go to zero as $n\to \infty$, the series must necessarily diverge. $$

<< Previous Example   Section 8.3    Next Example >>

© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.

For more information on Maplesoft products and services, visit www.maplesoft.com$$