Limit $\lim_{n \rightarrow \infty } \frac{1}{\sqrt{nn}} + \frac{1}{\sqrt{n(n+1)}}+ \frac{1}{\sqrt{n(n+2)}}+\cdots+ \frac{1}{\sqrt{n(n+n)}} .$

Question

The Question: Find $$\lim_{n \rightarrow \infty } \frac{1}{\sqrt{nn}} + \frac{1}{\sqrt{n(n+1)}}+ \frac{1}{\sqrt{n(n+2)}}+\cdots+ \frac{1}{\sqrt{n(n+n)}} .$$

The attempt:

I rewrote this as a series: $\displaystyle\sum_{k=0}^{n} \frac{1}{\sqrt{n(n+k)}}.$ Then I tried to represent this sum as a Riemann Sum. The width of the partitions is represented as the regular partitions:

$$\displaystyle\sum_{k=0}^{n} \frac{1}{\sqrt{n(n+k)}} = \sum_{k=0}^{n} \frac{\sqrt{n}}{n\sqrt{(n+k)}} = \sum_{k=0}^{n}\left(\frac{1}{n}\right)\sqrt{\frac{n}{n-k}}.$$

I am not sure where to go from here. Do you think I am on the right track?

Please I wants hints. Try not and solve the problem completely.

Thank you very much!

Answer 1

Now \begin{align*} \lim_{n\to \infty} \sum_{k=0}^{n} \frac{1}{n} \frac{1}{\sqrt{1+\frac{k}{n}}} &= \int_{0}^{1} \frac{dx}{\sqrt{1+x}} \\ &= \left[ 2\sqrt{1+x} \right]_{0}^{1} \\ &= 2(\sqrt{2}-1) \end{align*}