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July 26th, 2014, 12:21 PM  #1 
Senior Member Joined: Jul 2014 From: united states Posts: 114 Thanks: 5  convergence/conditional convergence/ divergence
determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum_{n=1}^{\infty} (\frac{n^2+2}{3n^2+2})^n$ how to do this problem?? im thinking root test. but how would i tell if its convergent or conditionally convergent? 
July 26th, 2014, 12:51 PM  #2 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  root test
$\displaystyle \limsup_{n\to\infty}\sqrt[n]{\left\frac{n^2+2}{3n^2+2}\right^{n}}=\limsup_{n \to\infty}\left\frac{n^2+2}{3n^2+2}\right=\frac{ 1}{3}<1$, so the series is convergent. Another example for you to consider, take the series $\displaystyle \sum_{n=1}^{\infty}\frac{(1)^n}{n}<\infty$ (conditionally convergent), because it is not absolutely convergent $\displaystyle \sum_{n=1}^{\infty}\left\frac{(1)^n}{n}\right=\sum_{n=1}^{\infty}\frac{1}{n}= \infty$ (harmonic series). Last edited by ZardoZ; July 26th, 2014 at 01:11 PM. 
July 26th, 2014, 02:38 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,497 Thanks: 580 
General comment: If all the terms of a series are nonnegative, then series is either divergent or absolutely convergent. 

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