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$$\lim_{n\to\infty}\sum_{k=1}^{n}{{1}\over{\sqrt{n^2+2kn}}}$$

How should one approach such a question? A hint would be helpful

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Check the question , It is very similiar math.stackexchange.com/questions/266322/… –  Mathlover Jan 28 '13 at 9:00
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Squeeze DOES work here. –  Did Feb 12 '13 at 10:11
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2 Answers

up vote 4 down vote accepted

Hint: rewrite this as a Riemann sum, so that the limit is an integral.

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I try
\begin{align}\lim_{n\to\infty}\sum_{k=1}^{n}{{1}\over{\sqrt{n^2+2kn}}}&=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}{{1}\over{\sqrt{1+\frac{2k}{n}}}}\\ &=\int_{0}^1\frac{1}{\sqrt{1+2x}}\,\,dx\end{align}

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