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July 26th, 2014, 12:21 PM   #1
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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?
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July 26th, 2014, 12:51 PM   #2
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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).
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Last edited by ZardoZ; July 26th, 2014 at 01:11 PM.
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July 26th, 2014, 02:38 PM   #3
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General comment:

If all the terms of a series are non-negative, then series is either divergent or absolutely convergent.
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