how to find the limit of sum here? Can it be transferred to an integral?

Question

$$\lim_{n\rightarrow\infty}\sum\limits_{k=1}^n \frac{k}{k^2+n^2}$$

I was trying to write it as an integral but failed, how to do it here?

Answer 1

We can write

$$\sum_{k=1}^n \frac{k}{k^2+n^2}=\sum_{k=1}^n \frac{(k/n)}{1+(k/n)^2}\frac1n$$

Taking the limit as $n\to \infty$ yields the Riemann sum for $\frac{x}{1+x^2}$ from $0$ to $1$. Therefore,

$$\lim_{n\to \infty}\sum_{k=1}^n \frac{k/n}{1+(k/n)^2}\frac1n=\int_0^1 \frac{x}{1+x^2}\,dx=\frac12 \log(2)$$