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Kempner Series


dsumOEIS
023.10344A082839
116.17696A082830
219.25735A082831
320.56987A082832
421.32746A082833
521.83460A082834
622.20559A082835
722.49347A082836
822.72636A082837
922.92067A082838

A Kempner series K_d is a series obtained by removing all terms containing a single digit d from the harmonic series. Surprisingly, while the harmonic series diverges, all 10 Kempner series converge. For example,

 K_1=1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/(20)+1/(22)+1/(23)+....

While they are difficult to calculate, the above table summarizes their approximate values as computed by Baillie (1979; Havil 2003, pp. 33-34).

Schmelzer and Baillie (2008) have devised an improved algorithm for summing more general Kempner series, such as the sum of sum_(n=1)^(infty)1/n where the digits of n contain no string 314. This sum has approximate value 2299.829782.... In general, the sum_(k=1)^(infty)1/k when a particular string of length n is excluded from the k's summed over is approximately given by 10^nln10 (Baillie and Schmelzer 2008).


See also

Harmonic Series

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References

Baillie, R. "Sums of Reciprocals of Integers Missing a Given Digit." Amer. Math. Monthly 86, 372-374, 1979.Baillie, R. and Schmelzer, T. "Summing Kempner's Curious (Slowly-Convergent) Series." May 20, 2008. http://library.wolfram.com/infocenter/MathSource/7166/.Havil, J. "The Kempner Series." §3.3 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 31-34, 2003.Schmelzer, T. and Baillie, R. "Summing Kempner's Curious, Slowly-Convergent Series." Amer. Math. Monthly 115, 525-540, 2008.Sloane, N. J. A. Sequences A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839 in "The On-Line Encyclopedia of Integer Sequences."Wadhwa, A. D. "Some Convergent Subseries of the Harmonic Series." Amer. Math. Monthly 85, 661-663, 1978.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers, rev. ed. Middlesex, England: Penguin Books, 1997.

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Kempner Series

Cite this as:

Weisstein, Eric W. "Kempner Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KempnerSeries.html

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