Limit $ \lim_{n \to \infty} \sum_{k=1}^n\frac 1 { \sqrt{n^2+k} } $ (1)
$$ \lim_{n \to \infty} \sum_{k=1}^n \frac 1 { \sqrt{n^2+k} }    $$
(2)
$$ \lim_{n\to\infty} \frac {1+\sqrt[n]2 + \sqrt[n]3 + ... \sqrt[n]n} {n}  $$
The answers should both be 1.. any hints?

For the first problem, note that if $1\le k\le n$, then 
$$n^2\lt n^2+k\le n^2+n\lt\left(n+\frac{1}{2}\right)^2.$$
Thus 
$$\frac{1}{n+\frac{1}{2}}\le \frac{1}{\sqrt{n^2+k}}\lt \frac{1}{n}.$$
Our sum is therefore between $\frac{n}{n+\frac{1}{2}}$ and $1$. 
Squeeze.
