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How to evaluate this limit?

$$\underset{n\to \infty }{\mathop{\lim }}\,\left( {{n}^{-2}}\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{n}^{2}}}{\frac{1}{\sqrt{{{n}^{2}}+ni+j}}}} \right)$$


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1 Answer 1

up vote 3 down vote accepted

The idea here is to rearrange in terms of a Riemann sum that leads to a double integral. Here, a little manipulation produces

$$\frac{1}{n} \sum_{i=1}^n \frac{1}{n^2} \sum_{j=1}^{n^2} \frac{1}{\sqrt{1+\frac{i}{n}+\frac{j}{n^2}}}$$

Then as $n \to \infty$, the double sum becomes

$$\begin{align}\int_0^1 dx \, \int_0^1 dy \frac{1}{\sqrt{1+x+y}} &= 2 \int_0^1 dx \left [ \sqrt{2+x}-\sqrt{1+x}\right ]\\ &= \frac{4}{3} \left [\left (3^{3/2}-2^{3/2} \right ) - \left (2^{3/2}-1 \right )\right ]\\&=4 \sqrt{3} - \frac{16}{3} \sqrt{2}+1\end{align}$$

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Nice solution, but I had preferred it if you had taken the homework tag into account. – Lord_Farin May 27 '13 at 11:50
@Lord_Farin: yeah, well, I got a little enthusiastic. Cat's out of the bag, I guess. – Ron Gordon May 27 '13 at 11:51
Thank you very much! – Sleepingip May 27 '13 at 12:59

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