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

Question

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

Is there some sort of a theorem or a method behind this type of limits? I mean, I can't even begin to do the task, since I have no clue. Recent findings have shown that it might be related to Rieman's sums, yet again, this hardly makes the matters clear.

See https://math.stackexchange.com/questions/1105719/evaluating-the-limit-lim-n-to-infty-left-fracnn21-fracnn222?noredirect=1&lq=1 — lab bhattacharjee, Jul 10 '17 at 11:13
Have you tried writing this as $$\lim_{n\to\infty} \left(\sum_{k=1}^{n} \frac{n}{n^2+k^2}\right)$$ — lioness99a, Jul 10 '17 at 11:16

score 0 · Answer 1 · answered Jul 10 '17 at 11:22

Rewriting the equation as $$S=\lim_{ n\rightarrow \infty }{ \sum_{r=1}^{n}{ \frac { n }{ { n }^{ 2 }+{ r }^{ 2 } } } } $$ $$S=\lim _{ n\rightarrow \infty }{ \frac { 1 }{ n } \sum _{ r=1 }^{ n } \frac { 1 }{ 1+{ \left( \frac { r }{ n } \right) }^{ 2 } } } $$ Converting Summation to Integration

$$S\quad =\quad \int _{ 0 }^{ 1 }{ \frac { 1 }{ 1+{ x }^{ 2 } } } dx$$ $$S=\arctan { (1) } -\arctan { (0) } =\frac { \pi }{ 4 } $$

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

1 Answers1