T0 find the value of $\lim\limits_{n\to\infty}\sum\limits_{k=1}^{n}\frac{1}{\sqrt{n^{2}+kn}}$

Question

What is $$\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{\sqrt{n^{2}+kn}}$$.I try to check whether series can be written as telescoping series but it doesn't work. I am getting no idea how to start?

Answer 1

Write it as $$\sum_{k=1}^n \frac{1}{n} \frac{1}{\sqrt{1+k/n}} $$ and interpret this as a Riemann sum for an integral.

Answer 2

By Riemann sum we have $$\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{\sqrt{n^{2}+kn}}=\displaystyle\lim_{n\to\infty}\frac1n\sum_{k=1}^{n}\frac{1}{\sqrt{1+k/n}}\to \int_0^1\frac{dx}{\sqrt{1+x}}= 2(\sqrt2-1)$$