{"paper":{"title":"Explicit Evaluations of Sums of Sequence Tails","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ce Xu, Xiaolan Zhou","submitted_at":"2017-01-04T04:36:23Z","abstract_excerpt":"In this paper, we use Abel's summation formula to evaluate several quadratic and cubic sums of the form: \\[{F_N}\\left( {A,B;x} \\right) := \\sum\\limits_{n = 1}^N {\\left( {A - {A_n}} \\right)\\left( {B - {B_n}} \\right){x^n}} ,\\;x \\in [ - 1,1]\\] and \\[F\\left( {A,B,\\zeta (r)} \\right): = \\sum\\limits_{n = 1}^\\infty {\\left( {A - {A_n}} \\right)\\left( {B - {B_n}} \\right)\\left( {\\zeta \\left( r \\right) - {\\zeta_n}\\left( r \\right)} \\right)} ,\\] where the sequences $A_n,B_n$ are defined by the finite sums ${A_n} := \\sum\\limits_{k = 1}^n {{a_k}} ,\\ {B_n} := \\sum\\limits_{k = 1}^n {{b_k}}\\ ( {a_k},{b_k} =o(n^{-p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.03725","kind":"arxiv","version":2},"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"}