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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.

StubbornAtom
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Barbara
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1 Answers1

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$$\begin{align}\lim_{n\to\infty}\sum_{1\le r\le n}\frac{n}{n^2+r^2} &=\lim_{n\to\infty}\frac1n\sum_{1\le r\le n}\frac1{1+\left(\frac rn\right)^2}\\ &=\int_0^1\frac{dx}{1+x^2}\end{align}$$

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

ParaH2
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lab bhattacharjee
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  • Very elegant method. Thank you! – Barbara Aug 17 '13 at 16:58
  • @Barbara, my pleasure. Related : http://math.stackexchange.com/questions/465075/find-lim-limits-n-to-infty-frac1n-sum-limits2n-r-1-fracr-sq. I'm really eager to about the alternatively methods, if any. – lab bhattacharjee Aug 17 '13 at 16:59
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    @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 '14 at 09:14
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    Wow, that is a beautiful formulae. At first it made no sense to me. But the more I look at it, the more 'obvious' it seems. $\sum_{r=1}^n$ basically adds all points from $r=1$ to $n$. But $f\left(\frac{r}{n}\right)$ divides $r$ by $n$. Which means the highest value is $1$. When $\lim_{n \to\infty}$ is applied, two things will happen: 1) Ratio $\frac{r}n$ approaches $0$. 2) Infinitesimal increment of the ratio at every step. But what is the other way to describe the summation of infinite no. of points from 0 to 1 of a function?$$\int_{0}^{1} f(x)dx$$ – Macindows Jan 30 '17 at 05:42
  • @lab bhattacharjee Can you please provide a general rigourous proof of this formula as a link or in some reference book and also its general form ? – Matt May 15 '17 at 07:51
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    @RaghavSingal, See http://www.askiitians.com/iit-jee-definite-integral/definite-integral-as-limit-of-a-sum/ – lab bhattacharjee May 15 '17 at 08:11
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    @RaghavSingal, Can you google please? – lab bhattacharjee May 18 '17 at 06:40
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    @RaghavSingal, See http://people.math.sc.edu/sharpley/math554_s96/554_9/ OR https://math.stackexchange.com/questions/2137518/proof-for-the-riemann-integral-in-terms-of-riemann-sums-using-the-epsilon-delta – lab bhattacharjee May 18 '17 at 06:57