Find a reordering of the alternating harmonic sequence such that the limit of the partial sums is $\infty$.

Question

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.

Answer 1

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.

Answer 2

Yes, the strategy is the following (slight modification of the usual one).

1. Let any $l_n$ (stricly) increasing sequence tending to infinity.
2. Take enough positive terms that their sum exceeds $l_1$ and stop
3. Take enough negative terms in order (by addition) to return before $l_1$ and stop
4. Take again enough positive terms in order to exceed $l_2$ and stop
5. Take enough negative terms in order (by addition) to return before $l_2$ and stop
6. ...

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