# If $a_n > 0$ and $\lim na_n = \ell$ with $\ell \ne 0$, then the series $\sum a_n$ diverges

If $a_n > 0$ and $\lim na_n = \ell$ with $\ell \ne 0$, then the series $\sum a_n$ diverges.

I've tried proving this using the root and ratio tests but to no avail.

Since $\lim\limits_{n\to\infty}na_{n}=l$ so there is $N$ such that for $n>N$ you have $$|na_{n}-l|<\epsilon$$ This says $$na_{n}>l-\epsilon$$ choose your $\epsilon=\frac{l}{2}$ and you have $na_{n}>\frac{l}{2}$ for $n>N$. Now note that $\sum_{n>N}\frac{l}{2n}$ diverges and use comparison test.