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. \ud \ud The first author is supported by a Rhodes Scholarship
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International audienceWe prove a bound for quintilinear sums of Kloosterman sums, with congruence co...
In this note, we extend a result of Sofo and Hassani concerning the evaluation of a certain type of ...
AbstractIn this paper we consider five conjectured harmonic number identities similar to those arisi...
The harmonic series diverges. But if we delete from the harmonic series all terms whose denominators...
We consider what value the harmonic series will converge to if evaluated in the obvious way using s...
AbstractTwo open problems recently proposed by Xi and Luo (Adv. Differ. Equ. 2021:38, 2021) are reso...
AbstractWe present a method for calculating any (nested) harmonic sum to arbitrary accuracy for all ...
This article describes the discovery and the subsequent proof of a hypothesis concerning harmonic se...
A historical development of the harmonic series subseries that are convergent is made. It is well kn...
AbstractAfter having recalled some important results about combinatorics on words, like the existenc...
AbstractQuite often in application, logarithmically convergent series have to be evaluated. There ar...
The paper deals with the reduced harmonic series generated by four primes. A formula for the sum of ...
In joint work Robin Pemantle and I (2004) consider a doubly infinite sum which is not equal to 1, as...
AbstractLet Sn denote the nth partial sum of the harmonic series. An asymptotic approximation for Sn...
This paper deals with certain generalization of the alternating harmonic series ? the generalized co...
International audienceWe prove a bound for quintilinear sums of Kloosterman sums, with congruence co...
In this note, we extend a result of Sofo and Hassani concerning the evaluation of a certain type of ...
AbstractIn this paper we consider five conjectured harmonic number identities similar to those arisi...