Kempner series

The harmonic series

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

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.