# Kempner series

 $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}\;=\;\frac{1}{1}+\frac{1}{2}+\frac{% 1}{3}+\frac{1}{4}+\ldots$ (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 $0,\,1,\,2,\,\ldots,\,8$ and for digit strings, as “716”.  E.g., we show the convergence of the partial series

 $\displaystyle K_{0}\;\begin{cases}=&\!\!\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+% \frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}\\ +&\!\!\frac{1}{11}+\frac{1}{12}+\ldots+\frac{1}{19}+\frac{1}{21}+\ldots\ldots+% \frac{1}{99}\\ +&\!\!\frac{1}{111}+\frac{1}{112}+\ldots+\frac{1}{119}+\frac{1}{121}+\ldots% \ldots+\frac{1}{999}\\ +&.\;.\;.\;.\;.\;.\end{cases}$

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

 $K_{0}\;<\;9\!\cdot\!\frac{1}{1}+9\!\cdot\!9\cdot\!\frac{1}{10}+9\!\cdot\!9\!% \cdot\!9\!\cdot\frac{1}{100}+\ldots\;=\;9+9\!\cdot\!\frac{9}{10}+9\!\cdot\!% \left(\frac{9}{10}\right)^{\!2}+\ldots\;=\;\frac{9}{1\!-\!\frac{9}{10}}\;=\;90$

(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 KempnerSeries 2013-03-22 19:12:59 2013-03-22 19:12:59 pahio (2872) pahio (2872) 8 pahio (2872) Result msc 40A05 depleted harmonic series