Kempner series


The harmonic series

n=11n=11+12+13+14+ (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 convergentMathworldPlanetmath 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,, 8 and for digit strings, as “716”.  E.g., we show the convergence of the partial series

K0{=11+12+13+14+15+16+17+18+19+111+112++119+121++199+1111+1112++1119+1121++1999+......

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

K0< 911+99110+9991100+= 9+9910+9(910)2+=91-910= 90

(a sum of convergent geometric seriesMathworldPlanetmath).

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