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