Convergence Rates of Subseries
Paolo Leonetti · arXiv (Cornell University) · 2018
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 $θ\in (0,1)$ such that, for each $\ell>0$, there is a subsequence $(x_{n_k})$ for which $\sum_k x_{n_k}=\ell$ and $x_{n_k}=O(θ^k)$.