Example of a rearrangement that diverges I know that any conditional convergent series has a rearrangement that diverges. For example if we have 
$$
\sum_{n=1}^\infty \frac{(-1)^n}{ n}
$$
what is a rearrangement that diverges?
 A: Not sure if it helps but:
$$\sum_{n=1}^\infty \frac{(-1)^n}{n} = \underbrace{\sum_{n=1}^\infty \frac{1}{2n}}_{S_1} - \underbrace{\sum_{n=1}^\infty \frac{1}{2n-1}}_{S_2}$$
But $S_1$ and $S_2$ diverge.
A: Let $S=\sum\limits_{n=1}^\infty (-1)^n a_n$, $a_n\ge0$, be conditionally convergent.
To obtain a divergent rearrangement of $S$, you can take advantage of the fact  that the sum of the positive terms of $S$ diverges to $\infty$. Then, given any $N$, you can find an $M $  so that the sum of successive positive terms, 
$a_{2N}+a_{2(N+1)}+\cdots+ a_{2( N+M)}$ is as large as you like.
You then construct the rearrangement by taking the sum of successive positive terms until you exceed $1$ (or any fixed positive number). Then add the first negative term. Then add the next block of successive positive terms whose sum exceeds $1$. Then add the next negative term. And so on.
You need to be careful to exhaust all terms of the series (so that you indeed have a rearrangement). If you do, then the resulting rearrangement will diverge as the resulting sequence of partial sums will not be Cauchy.
One can write an explicit example of a divergent rearrangement (as I attempted to do in the comments for your example).
