Does $a_{n} = \frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + ... + \frac{1}{\sqrt{n^2+2n-1}}$ converge? $a_{n} = \frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + ... +  \frac{1}{\sqrt{n^2+2n-1}}$
and I need to check whether this sequence converges to a limit without finding the limit itself. I think about using the squeeze theorem that converges to something (I suspect '$1$').
But I wrote $a_{n+1}$ and $a_{n-1}$ and it doesn't get me anywhere...

On the one hand, 
$$a_n \ge \mbox{smallest summand} \times \mbox{number of summands}= \frac{1}{\sqrt{n^2+2n-1}}\times  n .$$
To deal with the denominator, observe that  
$$n^2+2n-1 \le n^2+2n+1=(n+1)^2.$$
On the other hand, 
$$a_n \le \mbox{largest  summand}\times \mbox{number of summands} = \frac{1}{\sqrt{n^2+n}}\times n.$$ 
To deal with the denominator, observe that 
$$n^2+n \ge n^2$$. 
