A subseries of the series $\displaystyle\sum _{n=1}^\infty a_n$ is defined to be a series of the form $\displaystyle\sum _{k=1}^\infty a_{n_k}$, for $n_k \subseteq \Bbb N$.

**Prove or disprove:** For any conditionally convergent series $\displaystyle\sum _{n=1}^\infty a_n,\ \exists\ k\geq 2\ $ such that the subseries $\displaystyle\sum _{n=1}^\infty a_{kn}$ converges.

Certainly, every conditionally convergent series has some subseries that converge (decreasing alternating terms), and some subseries that diverge (the subseries of all of the positive terms; also the subseries of all of the negative terms).

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My original "answer" which I now doubt is a counter-example to the proposition:

Hint: Find a subset $\ A \subset \mathbb{N}\ $ with the property that, for each $\ n \in \mathbb{N},\ $ the set $ \{kn\}_{k \in \mathbb{N} }$ has finitely many elements in common with $A.$

Answer: The prime numbers is the standard example of such a subset of $\mathbb{N}$ with the property in the hint. This point is to make the 2nd, 3rd, 5th, 7th, ... members of our sequence have one sign, and every other member of our sequence have another sign. For example, $$ a_1 = \frac{1}{1},\ a_2 = \frac{-1}{4},\ a_3 = \frac{-1}{4},\ a_4 = \frac{1}{3},\ a_5 = \frac{-1}{4},\ a_6 = \frac{1}{5},\ a_7 = \frac{-1}{6},\ a_8 = a_9 = a_{10} = \frac{1}{3} \times \frac{1}{7},\ a_{11} = \frac{-1}{8},\ ... $$ This is simply the alternating harmonic series split up in a way that (I thought) answers the question.

That *was* my answer, but now I'm not sure the series above is a valid counter-example. I constructed it so that every subseries of the form $\displaystyle\sum _{n=1}^\infty a_{kn}$ contains finitely many negative numbers. And then I'm not sure what my logic was. It was either that "every subseries of a monotonic divergent series diverges", but now I realise that this is not what the link says, and is also not true. Or my reasoning was that, for each $k \in \mathbb{N},\ $ every subseries $\displaystyle\sum _{n=1}^\infty \frac{1}{kn}$ of the harmonic series, diverges, which is true, but the subseries of my series defined above do not match these subseries of the harmonic series.

So the problem remains open...