$\lim_{n \rightarrow \infty} \sum_{k=0}^n \left(\frac{1}{\sqrt {(n+k)(n+k+1)}}\right)$

Question

Hello everyone I have this problem

$$\lim_{n \rightarrow \infty} \sum_{k=0}^n \left(\frac{1}{\sqrt {(n+k)(n+k+1)}}\right)$$

I try to find some function to squeeze this but it seems that it does not work ( maybe I do it wrong? ).

Also, i try to simplify the function inside the sum too but I don't know what to process next.

$$\left(\frac{1}{\sqrt {(n+k)(n+k+1)}}\right)=\left(\frac{\sqrt{n+k+1}}{\sqrt {n+k}}\right)-\left(\frac{\sqrt{n+k}}{\sqrt {n+k+1}}\right)$$

Please help. Thank you very much!

Answer 1

$$S=\lim_{n \rightarrow \infty} \sum_{k=0}^n \left(\frac{1}{\sqrt {(n+k)(n+k+1)}}\right)\sim \lim_{n \rightarrow \infty} \sum_{k=0}^n \left(\frac{1}{n+k}\right).$$ $$S= \lim_{n \rightarrow \infty} \sum_{k=0}^n \frac{1}{n}\left(\frac{1}{1+k/n}\right)=\int_{0}^{1} \frac{dx}{1+x}=\log 2.$$