Does there exist a sequence $x_n$ such that $ \sum x_n >0$ but $ \sum x_n ^3 <0 $ where $ |x_n|<1$ Does there exist a sequence $x_n$ such that $ \sum_{n=1}^{\infty} x_n >0$ but $ \sum_{n=1}^{\infty} x_n ^3 <0 $ where $ |x_n|<1$ for all $n$?
Intuitively I would say no, since cubing preserves the sign of each term and since the first sum is greater than $0$, we can match up each -ve term with a sufficient number of +ve terms such that their sum is +ve. So after cubing each of these 'partial sums' will have the same sign (i.e. also +ve) hence the overall sum is also positive.
Is this correct? Is there a better way of wording the correct argument?

Take, for instance, $x_n=-3/5$ if $n$ is divisible by $3$ and $2/5$ otherwise. Then $\sum_{n=1}^\infty x_n=\infty$ while $\sum_{n=1}^\infty x_n^3=-\infty$.
