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How to compute this limit:

$$\lim_{n\to\infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots+\frac{n}{n^2+n^2}\right)$$

Please give me some hint. Thank you.

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1 Answer 1

$$\lim_{n\to\infty}\sum_{1\le r\le n}\frac{n}{n^2+r^2}$$

$$=\lim_{n\to\infty}\frac1n\sum_{1\le r\le n}\frac1{1+\left(\frac rn\right)^2}$$

$$=\int_0^1\frac{dx}{1+x^2}$$

$$\text{as }\lim_{n \to \infty} \frac1n\sum_{r=1}^n f\left(\frac rn\right)=\int_0^1f(x)dx$$

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Very elegant method. Thank you! –  Barbara Aug 17 '13 at 16:58
    
@Barbara, my pleasure. Related : math.stackexchange.com/questions/465075/…. I'm really eager to about the alternatively methods, if any. –  lab bhattacharjee Aug 17 '13 at 16:59
    
@lab bhattacharjee. You provided, for sure, the best and simplest solution to the problem. What amazed me was to have a look to the sum up to $n$. What I got (using a CAS) is $$\frac{-i n \left(H_{(1-i) n}-H_{(1+i) n}\right)+\pi n \coth (\pi n)-1}{2 n}$$ which goes to the limit you gave (fortunately for me !). Cheers. –  Claude Leibovici Jul 19 at 9:14

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