# Converse of alternating series test

Is the converse of the alternating series test true? In other words, given a sequence $a_n>0$, with neither $a_{2n}$ nor $a_{2n-1}$ constant, for which there exists no positive $N$ such that $a_n>a_{n+1}$ for all $n>N$, does $\displaystyle \sum_{k=1}^\infty (-1)^k a_k$ necessarily diverge? I've been unable to come up with any counterexamples, but I also can't figure out how to prove it.

No, just have the even terms be $1/n^2$ and the odd terms all be zero - that's not eventually monotone and the alternating sum converges.
• Fair enough; what if $a_k>0$? Your example is non-monotone, but the even and odd terms, taken separately, are both monotone, and one is a constant. I'll edit the question to be more clear.
• $a_{2n} = 1/n^2$, $a_{2n+1} = 1/n^3$ has convergent alternating sum and fits the hypotheses of the edited question. I don't think there will be any meaningful converse to the alternating series test - if the alternating sum diverges, then in particular the sum is not absolutely convergent, so the hypotheses would have to also imply the series not being absolutely convergent. Sep 24, 2014 at 10:58