the limit of infinite sums using stolz cesaro theorem

Question

I need to find the limit as n goes to infinity of

$$ {1\over\sqrt{n^2+1}}+{1\over\sqrt{n^2+2}}+\cdots+{1\over\sqrt{n^2+n+1}}. $$

I'm trying to figure out if you can use shtolz theorem.

Answer 1

In this case Stolz is a bad idea (you don't have a quotient), but with the squeeze theorem is esay: $$ {n+1\over\sqrt{n^2+n+1}}\le {1\over\sqrt{n^2+1}}+{1\over\sqrt{n^2+2}}+\cdots+{1\over\sqrt{n^2+n+1}}\le {n+1\over\sqrt{n^2+1}}. $$