On conditionally convergent series

Анотація

The most interesting result of the paper is that for any two complemen- tary subsets $A$ and $B$ of the set of positive odd integers there exists such a sequence $\{\alpha_k\}_{k=1}^{\infty}\subset [-1,1]$ that
$$ \forall m \in A :\textrm{the series } \displaystyle\sum_{k=1}^\infty \alpha_k^{m} \textrm{ is convergent and}$$
$$ \forall m \in B :\textrm{the series } \displaystyle\sum_{k=1}^\infty \alpha_k^{m} \textrm{ is divergent.}$$
Using the map $\overrightarrow{x}\longmapsto ||\overrightarrow{x}||^{\lambda}\frac{\overrightarrow{x}}{||\overrightarrow{x}||}$ as a substitute of the power function, one can prove similar results for vectors and positive not necessarily integer exponents $\lambda$.