Find a reordering of the alternating harmonic sequence such that the limit of the partial sums is $\infty$. The alternating harmonic series is defined as normal ($a_k = \frac{(-1)^{k+1}}{k}$).
Not sure how I could do this, as I would have to include all the positive and negative parts, which has a converging sum.
 A: Assuming your sequence is $$1-1/2+1/3-1/4+......$$
Start adding positive terms  to pass $1+1/2.$ 
Then subtract $1/2$ so at this point you have passed $1$ and used one of your negatives.
Now add enough positive points to pass $2+1/4$ and subtract $1/4$ so at this point you have passed $2$ and used two of your negatives.
Now add enough positive points to pass $3+1/6$ and subtract $1/6$ so at this point you have passed $3$ and used three of your negatives.
Keep repeating the process and you have used all your negatives while you sum diverges to infinity.
A: Yes, the strategy is the following (slight modification of the usual one). 


*

* Let any $l_n$ (stricly) increasing sequence tending to infinity.  

* Take enough positive terms that their sum exceeds $l_1$ and stop

* Take enough negative terms in order (by addition) to return before $l_1$ and stop

* Take again enough positive terms in order to exceed $l_2$
and stop

* Take enough negative terms in order (by addition) to return before $l_2$ and stop

* ...


can you formalize this ? (if not, we can interact) 
