Compute the limit of the sequence ${\textstyle\sum_{k=1}^n}\frac1{\sqrt{n^2+k}}$

Question

I have to compute the limit of this sequence ${\textstyle\sum_{k=1}^n}\frac1{\sqrt{n^2+k}}$ as $n\rightarrow\infty$. First I was thinking about some Riemann sum and and forced the $n^{2}$ outside the square root but the function was not so pleasant.

Answer 1

How about squeezing ?

$$\frac{n}{\sqrt{n^2+n}}\leq \sum_{k=1}^n\frac1{\sqrt{n^2+k}}\leq \frac{n}{\sqrt{n^2+1}}$$

The outer terms both go to $1$.