{
  "id": "0806.4410",
  "version": "v4",
  "published": "2008-06-27T10:51:28.000Z",
  "updated": "2015-08-27T19:17:37.000Z",
  "title": "Summing the curious series of Kempner and Irwin",
  "authors": [
    "Robert Baillie"
  ],
  "comment": "17 pages + 56 pages of Mathematica code for both Kempner and Irwin sums",
  "categories": [
    "math.CA",
    "math.NT"
  ],
  "abstract": "In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum is about 22.92068. In 1916, Irwin proved that the sum of 1/n where n has at most a finite number of 9's is also a convergent series. We show how to compute sums of Irwins' series to high precision. For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9, is about 23.04428708074784831968. Another example: the sum of 1/n where n has exactly 100 zeros is about 10 ln(10) + 1.007x10^-197 ~ 23.02585; note that the first, and largest, term in this series is the tiny 1/googol.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-08-19T01:04:50.000Z",
      "comment": "17 pages + 27 pages of Mathematica code",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-08-27T19:17:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "40-01",
      "65B10"
    ],
    "keywords": [
      "curious series",
      "finite number",
      "high precision",
      "convergent series",
      "actual sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.4410B"
    }
  }
}