# Kempner series

The harmonic series

$\sum _{n=1}^{\mathrm{\infty}}}{\displaystyle \frac{1}{n}}={\displaystyle \frac{1}{1}}+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{4}}+\mathrm{\dots$ | (1) |

is divergent. The situation is different when one omits from this series all terms whose denominators contain in the decimal system (http://planetmath.org/PositionalNumberSystems) some digits 9. Kempner proved 1914 very simply that such a “*depleted harmonic series*” is convergent^{} and that its sum is less than 90. This series is called *Kempner series*. A better value of the sum with 15 decimals is 22.920676619264150.

The digit 9 here has no special status; one has for other digits $0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots},\mathrm{\hspace{0.17em}8}$ and for digit strings, as “716”. E.g., we show the convergence of the partial series

${K}_{0}\{\begin{array}{cc}=\hfill & \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}\hfill \\ +\hfill & \frac{1}{11}+\frac{1}{12}+\mathrm{\dots}+\frac{1}{19}+\frac{1}{21}+\mathrm{\dots}\mathrm{\dots}+\frac{1}{99}\hfill \\ +\hfill & \frac{1}{111}+\frac{1}{112}+\mathrm{\dots}+\frac{1}{119}+\frac{1}{121}+\mathrm{\dots}\mathrm{\dots}+\frac{1}{999}\hfill \\ +\hfill & ......\hfill \end{array}$ |

of (1) where the denominators contain no 0’s. Every digit in the denominators has nine possibilities. For this series we thus get the estimation

$$ |

(a sum of convergent geometric series^{}).

For determining more accurately the sums of depleted harmonic series (cf. http://en.wikipedia.org/wiki/Depleted_uraniumdepleted uranium), see the article http://arxiv.org/ftp/arxiv/papers/0806/0806.4410.pdfSumming the curious series of Kempner and Irwin of Baillie.

Title | Kempner series |
---|---|

Canonical name | KempnerSeries |

Date of creation | 2013-03-22 19:12:59 |

Last modified on | 2013-03-22 19:12:59 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Result |

Classification | msc 40A05 |

Synonym | depleted harmonic series |