Evaluate:
$\displaystyle
\lim_{n\to \infty}(\frac{1}{\sqrt {n^2}} +\frac{1}{\sqrt {n^2+1}} +\frac{1}{\sqrt{n^2+2}} +.........+\frac{1}{\sqrt{n^2+2n}})$
As each of the denominators get larger and larger, we see that each of those terms approach zero.
So, $\displaystyle
\lim_{n\to \infty}(\frac{1}{\sqrt {n^2}} +\frac{1}{\sqrt {n^2+1}} +\frac{1}{\sqrt{n^2+2}} +.........+\frac{1}{\sqrt{n^2+2n}})$ $\displaystyle = 0+0+0+\dots+0=\color{red}\boxed{0}$
Does this make sense?
--Chris