TITLE: Convergence of a sequence, $a_n=\sum_1^nn/(n^2+k)$
QUESTION [7 upvotes]: Let $ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}$ . I would like to know whether the given sequence converges. 
I see  that,
$ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}= \sum_{k=1}^{n} \frac{1}{n+\frac{k}{n}}.$ When $n$ gets sufficiently large the contribution by the $ \frac{k}{n} $ term is diminishing and $ a_{n} < \sum_{k=1}^{n} \frac{1}{n} = 1 $. 
Thank you.

REPLY [6 votes]: $$\frac{1}{n+1}\leq \frac{1}{n+\frac{k}n} \leq \frac{1}{n}$$
So $\frac{n}{n+1} \leq a_n \leq 1$
So $a_n\rightarrow 1$.