Convergence of $\sum_{n=1}^\infty \frac{a_n}{n}$ with $\lim(a_n)=0$. Is it true that if $(a_n)_{n=1}^\infty$ is any sequence of positive real numbers such that $$\lim_{n\to\infty}(a_n)=0$$
then,
$$\sum_{n=1}^\infty \frac{a_n}{n}$$
converges?
If yes, how to prove it?

The problem is equivalent to finding a positive increasing function $f(n)$ that approaches $+\infty$ slower than the harmonic series.  
Given such a function, choose $a_n$ so that  $\sum \frac{1}{na_n} = f(n)$.  Then $\sum \frac{1}{na_n}$ diverges.
Or, for simpler formulas, take a continuous $f(x)$ and choose $a(x)$ so that $\frac{1}{xa(x)} = f'(x)$. Then $\int \frac{1}{xa(x)}$ diverges and consequently so does $\sum \frac{1}{na_n}$.
The most common choice is $f(x) = \log \log x$.
