Why can't I use Divergence test instead of Limit comparison test?

Question

$\sum_{n=1}^{\infty}\frac{n}{n^{2}+1}$ or series an = (n/n^2+1) from $n = 0$ to $\infty$. Why can't I use the divergence test here and multiply top and bottom by $\frac{1}{n^{2}}$ and get the limit is equal to $0$. I know I have to use limit comparison test to get the limit equals $1$. Why doesn't the divergence test work here?

Answer 1

Just because $$\lim_{n\to\infty} a_n\to0$$ does not mean that $$\sum_{n=0}^\infty a_n$$ converges. It simply means that the limit test is inconclusive.