$$\lim_{n \to \infty} \sqrt{n}\left(1 - \sum_{k=1}^n \frac{1}{n + \sqrt{k}}\right)$$
Guys, please help to solve the limit. I have try transfer it to integrate problem, but it works not good. My friends try to make the sum $f(n)$, it works but I don't know whether it done. I put it to python procedure getting the answer $2/3$. But I don't have specific math procedure. Thanks for help.
$$A_n:=\sqrt{n}\left(1-\sum_{k=1}^n\frac1{n+\sqrt{k}}\right)=\sqrt{n}\sum_{k=1}^n\left(\frac1n-\frac1{n+\sqrt{k}}\right)=\frac1n\sum_{k=1}^n\frac{\sqrt{k/n}}{1+\sqrt{k}/n}.$$
Since $1\leqslant 1+\sqrt{k}/n\leqslant 1+1/\sqrt{n}$, we get $$\frac{B_n}{1+1/\sqrt{n}}\leqslant A_n\leqslant B_n,\qquad B_n:=\frac1n\sum_{k=1}^n\sqrt{k/n}.$$
Hence $\lim\limits_{n\to\infty}A_n=\lim\limits_{n\to\infty}B_n$. Now $B_n$ is a Riemann sum for $\int_0^1\sqrt{x}\,dx=2/3$.