Let $$x_n=\frac{1}{1+n^2}+\frac{2}{2+n^2}+\dots+\frac{n}{n+n^2}$$ be a sequence. Prove or disprove that the sequence converges.
Here's what I have done so far: I tried to find the lower and upper bounds for each term and found that $$\frac{1}{2}\le x_n \le \frac{1}{2n}+\frac{1}{2}; \forall n\in \mathbb{N}$$ I was hoping to show that it is a monotone sequence and bounded. Any hints without the use of integrals are much appreciated.