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 ${\mathrm{liminf}}_n{x}_{n+1}/x_n>1/2$. 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 $x_{n_k}=O({\theta }^k)$.

5 pp. To appear in The American Mathematical Monthly