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$\begingroup$

So I need to know the convergence of this sum. The integral test does not seem to work (WolframAlpha gives a confusing answer which implies it's not the correct method). I also tried the limit comparison test for various harmonic series, but they are all inconclusive. I'm also not getting very far with the ratio test. $$ \sum_{n=2}^\infty \dfrac{\sqrt{n+1}}{n(n-1)} $$

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    Did you try a limit comparison test with $\displaystyle\sum \frac1{n^{3/2}}$? Your numerator is roughly $n^{1/2}$ and your denominator is roughly $n^2$, so your fraction is roughly $1/n^{3/2}$.2012-04-09
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    @BrianM.Scott: the fraction is less than $(n-1)^{-3/2}$ for $n\ge2$.2015-01-09

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