Limit of a nearly Riemann sum

Question

On of my student asked me to find $$\lim_{n\to+\infty}\sum_{k=1}^n\frac{1}{\sqrt{(n+k)(n+k+1)}}$$

I tried to write it as a Riemann sum, but the term $ \frac 1n $ disturbs.

I tried to use the fact that $$n+k\le n+k+1 \le 2n+1$$ but no way. Thank you in advance. appreciate.

Answer 1

$$\sum_{k=1}^n\frac{1}{n+k+1} \leq\sum_{k=1}^n\frac{1}{\sqrt{(n+k)(n+k+1)}}\leq\sum_{k=1}^n\frac{1}{n+k} $$

Let, $S_n=\sum_{k=1}^n\frac{1}{n+k}$

And $T_n=\sum_{k=1}^n\frac{1}{\sqrt{(n+k)(n+k+1)}}$

Then we have,

$$S_n- \frac{1}{n+1}+\frac{1}{2n+1}\leq T_n\leq S_n$$

Now use sandwich theorem.