{
  "id": "2011.10638",
  "version": "v1",
  "published": "2020-11-20T21:07:58.000Z",
  "updated": "2020-11-20T21:07:58.000Z",
  "title": "Convergent subseries of divergent series",
  "authors": [
    "Marek Balcerzak",
    "Paolo Leonetti"
  ],
  "comment": "6 pp; comments are welcome",
  "categories": [
    "math.CA",
    "math.FA",
    "math.GN"
  ],
  "abstract": "Let $\\mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $\\sum_n x_n$ is divergent. For each $x \\in \\mathscr{X}$, let $\\mathcal{I}_x$ be the collection of all $A\\subseteq \\mathbf{N}$ such that the subseries $\\sum_{n \\in A}x_n$ is convergent. Moreover, let $\\mathscr{A}$ be the set of sequences $x \\in \\mathscr{X}$ such that $\\lim_n x_n=0$ and $\\mathcal{I}_x\\neq \\mathcal{I}_y$ for all sequences $y=(y_n) \\in \\mathscr{X}$ with $\\liminf_n y_{n+1}/y_n>0$. We show that $\\mathscr{A}$ is comeager and that contains uncountably many sequences $x$ which generate pairwise nonisomorphic ideals $\\mathcal{I}_x$. This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-20T21:07:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "40A05",
      "54A20"
    ],
    "keywords": [
      "divergent series",
      "convergent subseries",
      "generate pairwise nonisomorphic ideals",
      "positive real sequences",
      "open question"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}