On conditionally convergent series

Vladimir Logvinenko

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2004) no. 4, pp. 470-483

Résumé

The most interesting result of the paper is that for any two complementary 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 \begin{gather*} \forall\,m\in A:\text{ the series }\sum_{k=1}^\infty\alpha_k^m\text{ is convergent and} \\ \forall\,m\in B:\text{ the series }\sum_{k=1}^\infty\alpha_k^m\text{ is divergent.} \end{gather*} 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 $\lambda$.