Define $u_n$ as :
$$u_n=\displaystyle\sum_{k=1}^n \dfrac{1}{\sqrt{n^2+k}}$$
From first values, $(u_n)$ seems to be increasing.
From squeezing, its limit is $1$.
$$ \dfrac{1}{\sqrt{n^2+n}} \le \dfrac{1}{\sqrt{n^2+k}} \le \dfrac{1}{\sqrt{n^2+1}}$$
(summing from $k=1$ to $k=n$)
How can one show that it's indeed increasing (if it is) because no terms cancel out when setting $u_{n+1}-u_n$.
thanks.
