Convergence Rates of Subseries

TL;DR

This paper investigates the convergence behavior of subseries of a divergent decreasing positive sequence, establishing conditions under which specific subseries converge to any positive sum with exponentially decreasing terms.

Contribution

It provides new conditions on the sequence that guarantee the existence of exponentially decreasing subseries summing to any positive value.

Findings

Existence of exponentially decreasing subseries for any positive sum

Conditions on the original sequence's ratio limit inferior

Subseries can be constructed with controlled decay rate

Abstract

Let $(x_n)$ be a positive real sequence decreasing to $0$ such that the series $\sum_n x_n$ is divergent and $\liminf_{n} x_{n+1}/x_n>1/2$. 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 $x_{n_k}=O(\theta^k)$.