How to compute $\lim\limits_{n\to\infty}\frac{1}{\sqrt{4n^2-1^2}}+\dots+\frac{1}{\sqrt{4n^2-n^2}}$. I want to compute $$\lim_{n\to\infty}\sum_{k=1}^n \frac1{\sqrt{4n^2-k^2}}.$$
I found it on this question and the exercise appears to be from previous years of a Latvian competition.
I tried writing $\frac1{\sqrt{4n^2-k^2}}=((2n-k)(2n+k))^\frac{-1}2$ and wanted to continue with partial fractions but the square root is annoying. Also, I thought about $$\frac1{\sqrt{4n^2-k^2}}=\frac{\sqrt{4n^2+k^2}}{\sqrt{16n^4-k^4}}$$ but I don't think that this can help me.

You can rewrite it to become $$\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n \frac{n}{\sqrt{4n^2-k^2}} = \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n \frac{1}{\sqrt{4-\left(\frac{k}{n}\right)^2}}$$
This is the Riemann sum for $\frac{1}{\sqrt{4-x^2}}$ from $0$ to $1$. Therefore, the limit of the sum is $$\int_0^1 \frac{1}{\sqrt{4-x^2}} dx$$
Integrating, we get $$\arcsin \left( \frac{x}{2} \right) \bigg\vert_0^1 = \arcsin\left(\frac{1}{2}\right) - \arcsin\left(\frac{0}{2} \right) = \frac{\pi}{6}$$
