Can this series by explicitly re-arranged not to converge? I need some help on a challenge problem from my analysis class this week. The question is whether the series
$s = \displaystyle \sum_{n \in \mathbb{N}} a_n = \sum \frac{(-1)^{n+1}}{n} =  1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} \ldots...$
Can be re-arranged in such a way so as not to converge? 
Chapter 3 of Principles of Mathematical Analysis by Rudin includes an example where the series can be re-arranged so as not to converge to the same value as $s$.
 A: Yes, it can. It can also be rearranged to converge to any specified real number. This is true of every conditionally convergent series.
To rearrange it to diverge to $\infty$, start by adding positive terms until the total is at least $1$; that takes just one term, $1$. Then add the first negative term that hasn’t yet been included; in this case you just get $1-\frac12$. That’s step $1$. Step $2$ is to add positive terms, starting where you left off in step $1$, until the partial sum is at least $2$, and then throw in the first unused negative term; you get
$$1-\frac12+\frac13+\frac15+\ldots+\frac1{41}-\frac14\;.$$
In general at step $n$ you add positive terms, starting where you left off in step $n-1$, until the partial sum is at least $n$, and then add the next negative term. 
This procedure can be carried out because the series $1+\frac13+\frac15+\frac17+\ldots$ diverges, so that at step $n$ we can always add enough positive terms to bring the partial sum up to $n$. And since the rearranged series has arbitrarily large partial sums, it must diverge.
