# Series $\sum a_n$ and $\sum a_n^3$ [duplicate]

Is possible to find series that: $\sum_{n=1}^\infty a_n$ is converges but $\sum_{n=1}^\infty a_n^3$ is diverges

I thought about something with $(-1)^n$ and Leibniz criterion but don't have idea.

## marked as duplicate by Andrés E. Caicedo, user61527, TZakrevskiy, Dietrich Burde, Hanul JeonNov 30 '13 at 0:20

Let $j=e^{2\pi i/3}$.
$\sum \cfrac{j^k}{k}$ converges but $\sum \cfrac{1}{k}$ diverges.
Maybe: $$\left(1,\frac{-1}{2} ,\frac{-1}{2} ,\frac{1}{\sqrt[3]{2}} ,\frac{1}{2}\cdot \frac{-1}{\sqrt[3]{2}} ,\frac{1}{2}\cdot \frac{-1}{\sqrt[3]{2}} ,..., \frac{1}{\sqrt[3]{n}} ,\frac{1}{2}\cdot \frac{-1}{\sqrt[3]{n}}, \frac{1}{2}\cdot \frac{-1}{\sqrt[3]{n}} ,\frac{1}{\sqrt[3]{n+1}} ,\frac{1}{2}\cdot \frac{-1}{\sqrt[3]{n+1}}, \frac{1}{2}\cdot \frac{-1}{\sqrt[3]{n+1}} ,...\right)$$