Limit and Sigma notation

Compute the following limit $$\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{\sqrt{n^2+kn}}$$

Please I need your help asap

Cheers

Matthew

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Any thoughts, efforts, ideas, insights...? –  DonAntonio Nov 6 '12 at 19:00
$\lim_{x \to \infty} \frac{1}{\sqrt{n}} \sum{\frac{1}{\sqrt{n+k}}}$ –  Matthew Nov 6 '12 at 19:04

1 Answer

$$\sum_{k=1}^n\frac{1}{\sqrt{n^2+kn}}=\frac{1}{n}\sum_{k=1}^n\frac{1}{\sqrt{1+\frac{k}{n}}}\xrightarrow [n\to\infty]{}\int_0^1\frac{dx}{\sqrt{1+x}}$$

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How did you get from step to step 3? –  Matthew Nov 6 '12 at 19:10
If you mean the partition, it is $\,\{1/n\,,\,2/n\,,...,\,n/n=1\}\,$ of the unit interval $\,[0,1]\,$ –  DonAntonio Nov 6 '12 at 19:11
@Matthew That step considers your sum as a Riemann Sum for the given integral. If this is something you're unfamiliar with, it's worth looking up. –  Daniel Littlewood Nov 6 '12 at 21:29