Convergence Rates of Subseries

Abstract

Let $(x_n)$ be a positive real sequence decreasing to 0 such that the series $\sum _n{x}_n$ is divergent and $\underset{n}{\lim \inf }{x}_{n+1}/x_n>1/2$. We show that there exists a constant $\theta \in (0,1)$ such that, for each $\ell >0$, there is a subsequence $(x_{n_k})$ for which $\sum _k{x}_{n_k}=\ell$ and $x_{n_k}=O({\theta }^k)$.

MSC:

• Primary 40A35

Acknowledgment

The author is grateful to the editor and the two anonymous referees for their remarks that allowed a substantial improvement of the presentation.