# 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.

## 2 Answers

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.

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)