An example of a series $\sum a_n$ that converges conditionally but $\sum a_n^3$ does not converge Give an example of a series $\sum a_n$ that converges conditionally but $\sum a_n^3$ does not converge conditionally.
I've come up with an example. 
$\frac{1}{\sqrt[3]2}-\frac{1}{2\sqrt[3]2}-\frac{1}{2\sqrt[3]2}+\frac{1}{\sqrt[3]3}-\frac{1}{3\sqrt[3]3}-\frac{1}{3\sqrt[3]3}-\frac{1}{3\sqrt[3]3}+\cdots$. 
While the sum of the cubes is 
$\frac{1}{2}-\frac{1}{8\cdot 2}-\frac{1}{8\cdot 2}+\frac{1}{3}-\frac{1}{27\cdot 3}-\frac{1}{27\cdot 3}-\frac{1}{27\cdot 3}+\cdots$
Now the series seems to converge to 0, however, I cannot show using an epsilon argument that it does. Also, the sum of the cubes looks like $\frac{1}{4}\cdot \frac{1}{2}+\frac{8}{9}\cdot \frac{1}{3}+ \frac{15}{16}\cdot \frac{1}{4}+\cdots$, so I can see that it diverges, but likewise, cannot supply this with a rigorous argument. 
I would greatly appreciate it if anyone can help me with this part. 

Consider , $a_n=\frac{(-1)^n}{n}$. Then $a_n$ is conditionally convergent. But , $a_n^3=\frac{(-1)^n}{n^3}$ is NOT conditionally  convergent ; as it is absolutely convergent.
