TITLE: Convergence of the sequence $\alpha_n = \frac{1}{\sqrt{n^2 + 1}} + \dots+\frac{1}{\sqrt{n^2+n}}$
QUESTION [6 upvotes]: How to determine the convergence of this sequence?
$$\alpha_n = \frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2+2}}+ \dots +\frac{1}{\sqrt{n^2+n}}$$
I was trying to show first that the sequence has a monotony with
$$\alpha_{n+1}-\alpha_n$$
but I couldn't figure out how.

REPLY [13 votes]: Use the sandwich theorem,
$$\frac{n}{\sqrt{n^2+n}}\leq \frac{1}{\sqrt{n^2+1}}+\cdots +\frac{1}{\sqrt{n^2+n}}\leq \frac{n}{\sqrt{n^2}}=1$$
it is then clear to see that the limit is $1$.