The Mathematical Institute, University of Oxford, Eprints Archive

Summing curious, slowly convergent, harmonic subseries

Schmelzer, Thomas and Baillie, Robert (2006) Summing curious, slowly convergent, harmonic subseries. Technical Report. Unspecified. (Submitted)



The harmonic series diverges. But if we delete from the harmonic series all terms whose denominators contain any string of digits such as "9", "42", or "314159", then the sum of the remaining terms converges. These series converge far too slowly to compute their sums directly. We describe an algorithm to compute these and related sums to high precision. For example, the sum of the series whose denominators contain no "314159" is approximately 2302582.33386. We explain why this sum is so close to 106 log 10 by developing asymptotic estimates for sums that omit strings of length n, as n approaches infinity.

The first author is supported by a Rhodes Scholarship.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1106
Deposited By: Lotti Ekert
Deposited On:11 May 2011 09:58
Last Modified:29 May 2015 18:47

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