Let $b_n$ be a decreasing positive sequence such that $\sum b_{2n}^3$ diverges to infinity. ($b_n=n^{-1/3}$ works.) The alternating series $\sum(-1)^nb_n$ and $\sum(-1)^nb_n^3$ must converge by the alternating series criterion.
Define the series $a_n$ to be:
\begin{align}
(a_n)=(&b_0,-b_1,\\
&b_2,-b_3/3,-b_3/3,-b_3/3\\
&b_4,-b_5/5,-b_5/5,-b_5/5,-b_5/5,-b_5/5\\
&\dots)
\end{align}
That is, replace $b_{2n+1}$ by a group of $2n+1$ copies of $b_{2n+1}/(2n+1)$.
$\sum a_n=\sum(-1)^nb_n$ by grouping the terms, and so it converges. I will show that $\sum a_n^3$ must diverge, solving the problem.
I realize this question has been posted but it is on hold. It piqued my interest and now I cant figure it out so Im looking for some help.
Question
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