Is it because it is comparable to $\sum_{n=1}^\infty \frac{1}{n}$ ?
2. Let be $a_{n}$ the general term of a convergent series. In this case is $\lim_{n \rightarrow \infty} a_{n}=0$ so that $\lim_{n \rightarrow \infty} \frac{1}{a_{n}} = \pm \infty$... that's why the series of the $\frac{1}{a_{n}}$ never converges ...
$\chi$ $\sigma$