{
  "id": "2009.14644",
  "version": "v1",
  "published": "2020-09-30T12:52:54.000Z",
  "updated": "2020-09-30T12:52:54.000Z",
  "title": "Irrationality and Transcendence of Alternating Series Via Continued Fractions",
  "authors": [
    "Jonathan Sondow"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Euler gave recipes for converting alternating series of two types, I and II, into equivalent continued fractions, i.e., ones whose convergents equal the partial sums. A condition we prove for irrationality of a continued fraction then allows easy proofs that $e,\\sin1$, and the primorial constant are irrational. Our main result is that, if a series of type II is equivalent to a simple continued fraction, then the sum is transcendental and its irrationality measure exceeds $2$. We construct all $\\aleph_0^{\\aleph_0}=\\mathfrak{c}$ such series and recover the transcendence of the Davison--Shallit and Cahen constants. Along the way, we mention $\\pi$, the golden ratio, Fermat, Fibonacci, and Liouville numbers, Sylvester's sequence, Pierce expansions, Mahler's method, Engel series, and theorems of Lambert, Sierpi\\'{n}ski, and Thue-Siegel-Roth. We also make three conjectures. (This manuscript was submitted posthumously. The author passed away on January 16, 2020.)",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-30T12:52:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J82",
      "11A55",
      "11B39"
    ],
    "keywords": [
      "alternating series",
      "transcendence",
      "euler gave recipes",
      "irrationality measure exceeds",
      "convergents equal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}