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- #1

- Apr 13, 2013

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I want to check if the sequence $a_{n}=\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+...+\frac{1}{\sqrt{n^2+n}}$ converges.

I thought that I could find the difference $a_{n+1}-a_{n}$ to check if $a_{n}$ is increasing or decreasing.I found:

$a_{n+1}-a_{n}=\sum_{i=1}^{n}(\frac{1}{\sqrt{(n+1)^{2}+i)}}-\frac{1}{\sqrt{n^2+i}})+\frac{1}{\sqrt{(n+1)^{2}+n+1}}$..But from that we cannot conclude if the difference is negative or positive,right?? So,what else could I do??