Value of $\lim_{n\to\infty}\sum_{k=1}^n \frac{n}{n^2+k^2}$

Question

$$\lim_{n\to\infty}\sum_{k=1}^n \frac{n}{n^2+k^2}$$

I tried this using powerseries just putting as $x=1$ there ,even tried thinking subtracting $s_{n+1} - s_{n}$ would be of some help, also I thought of writing the denominator as product of two complex numbers and then doing the partial fractions but it did not help. Any method guys?

Thanks in advance for guiding to think about these kind of problems.

Think about taking a certain Riemaninan Sum of a proper function on an interval. For example $f(x)=1/(1+x^2),~I=[0,1]$ — Mikasa, Jan 09 '16 at 04:14
You can probably find a few posts about this sum on this site. For example, The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$, How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$? or To show that the limit of the sequence $\sum\limits_{k=1}^n \frac{n}{n^2+k^2}$ is $\frac{\pi}{4}$. Found using Approach0. — Martin Sleziak, Jan 10 '17 at 17:52

score 1 · Accepted Answer · answered Jan 09 '16 at 04:14

1

Let $S(n)=\sum_{k=1}^{n}\frac{n}{n^2+k^2}$. We can evaluate the limit of this sum as a Riemann sum by writing

$$\begin{align} \lim_{n\to \infty}S(n)&=\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^{n}\frac{1}{1+(k/n)^2}\\\\ &=\int_0^1 \frac{1}{1+x^2}\,dx\\\\ &=\pi/4 \end{align}$$

answered Jan 09 '16 at 04:14

Mark Viola

179,405

Actually, I am unaware of Reimann Sums,kindly add a reference for a beginner .Is there any other method to solve this problem?Kindly suggest. – BAYMAX Jan 09 '16 at 07:30

score 1 · Answer 2 · answered Jan 09 '16 at 04:14

1

Hint : it is a riemann sum, just factorize $n$ to make $\frac{k}{n}$ appear

answered Jan 09 '16 at 04:14

stity

3,500

Value of $\lim_{n\to\infty}\sum_{k=1}^n \frac{n}{n^2+k^2}$

2 Answers2