This is in the section covering alternating sequences. Leibniz's rule, conditional/absolute convergence, Dirichlet's test, and Abel's tests were all covered.
I don't know what to apply here, it seems like none of the tests I have learned are applicable. Most of the theorems in this section covered sufficient conditions for convergence, which I can't turn back on themselves RAA to prove that this diverges. Obviously the series does not absolutely converge, and the even terms are greater in absolute value than the odd terms, so it will keep increasing, however I think there's a problem with approaching it this way since it is essentially grouping terms of the infinite sequence within parenthesis, which is not strictly allowed.
That the sequence most approach 0 as n increases without bound, but that is the case here.
Everything else which could prove divergence requires a positive series, so the only way I thought I could prove this would be to show that the sequence of partial sums is not bounded in some way. Of course, this summation does not have a nice summation formula.
The fact that the positive terms are greater (in absolute value) than their subsequent negative terms seems to indicate that , but obviously this alone is not enough to prove divergence.
The book did not even cover a theorem which would justify my approach here, it only made this statement for positive sequences. It doesn't seem hard to justify this, though, since if the sequence of partial sums is not bounded then assuming it converged would be a contradiction.
Let be the partial sum. Then
Taking the derivative yields
Which, for , is strictly greater than 0, and so the difference between partial sums is strictly increasing. Therefore, the series must diverge.
We haven't covered a specific theorem to reference here, but the basic definition of divergence at the end should be enough to arrive at that conclusion.
Does this look valid?
What you're essential showing is the terms in my series are not going to zero (nth term test). However, my question to you is how do you know this derivative is greater than zero for n greater or equation to 1?
EDIT. Actually the terms are going to zero. My bad.
this means that every other partial sum must also "eventually" be within epsilon of that same limit.
but if the subsequence of every other partial sum does not converge, how can the whole sequence possibly converge?
yes, "grouping" is illegal in establishing what the limit actually IS, and re-arranging terms is even worse. but that's not we are doing here.
my claim is: if a sequence converges, then any (infinite) subsequence must converge, as well. so if a subsequence does NOT converge, then the original sequence does not converge.
it is important that we keep the terms of the partial sums in the same order, no "skipping" terms, or moving them around.
to address the counter-example: 1 + -1 + 1 + -1 + 1 + -1 +......
here, the alternating subsequences and both converge, the former to 1, and the latter to 0.
this tells us nothing, to actually know something, we would have to know that the original sequence converges (so that we can deduce the convergence of the subsequences, and we don't know this), or we would have to know that one of the subsequences diverged (to deduce the divergence of the "parent sequence"), and that isn't the case here.