If an infinite series is divergent, after rearrangement can it be convergent?

Question

I was studying regarding Riemann rearrangement theorem which was regarding conditionally convergent series. Now I am wondering, if an infinite series is divergent, does there always exist some rearrangement of the series such that it is convergent?

Answer 1

Not necessarily. Consider the series $\displaystyle\sum_{n=1}^{\infty}a_n$ where $a_n = 1$ for every $n$.

Answer 2

Only if it's 'conditionally' divergent in the sense that the positive terms form a divergent series, and also the negative terms form a divergent series. You would also need $a_n\to 0$, of course. In this case, you can use the same algorithm for rearrangement in order to force convergence to some (arbitrary) value.