{"paper":{"title":"Convergence Rates of Subseries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Paolo Leonetti","submitted_at":"2018-05-25T21:11:23Z","abstract_excerpt":"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 $x_{n_k}=O(\\theta^k)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.10366","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}