Rearrangement of alternating harmonic series that does not converge From Riemann's series theorem, we know that, given a conditionally convergent series, we can permute the elements of the series in order to basically do whatever we want. I have seen a rearrangement of the alternating harmonic series that converges to $\frac{3}{2}\ln{2},\frac{1}{2}\ln{2},\infty$, but I have not seen arrangements that do not converge to anything, that is not even to $\pm\infty$. 
My idea was to somehow create an arrangement such that two subsequences of the partial sums that each will tend to a different value, but I was not able to come up with one.
 A: If $\sum a_n$ is a conditionally convergent series of real numbers and $\alpha\le\beta$ then there is a rearrangement so that $\liminf s_n=\alpha$ and $\limsup s_n=\beta$.
The proof is the same as the proof that there is a rearrangement converging to $\alpha$. Start with just enough positive terms to give a partial sum larger than $\beta$, then add just enough negative terms to give a partial sum less than $\alpha$, then add more positive terms til you get a partial sum larger than $\beta$ again, etc. See the proof of the rearrangement theorem that you mention for details.
A: Fix two real number $\alpha,\beta
 $ and assuming $\alpha>\beta
 $. Since $\sum_{n\geq1}\frac{1}{2n}
 $ diverges, we can find some $N_{1}
 $ such that $$\alpha<\sum_{n\leq N_{1}}\frac{1}{2n}:=S
 $$ and now since $\sum_{n\geq1}\frac{1}{2n+1}
 $ diverges we can find some $N_{2}
 $ such that $$\beta>S-\sum_{n \leq N_{2}}\frac{1}{2n+1}
 $$ and so on. So $\alpha,\beta
 $ are accumulation point of the partial sums.
A: In fact, if $\sum a_n$ is conditionally convergent, then there is a rearrangement of $\sum a_n$ such that the partial sums of the rearranged series form a dense subset of $\mathbb R.$ 
