Let Sn denote the nth partial sum of the harmonic series. For a given positive integer k>1, there exists a unique integer nk such that Snk-1<Snk. It has been conjectured that nki is equal to the integer nearest ek-γ, where γ is Euler's constant. We provide an estimate on nk which suggests that this conjecture may have to be modified. We also propose a conjecture concerning the amount by which Snk- and Snk differ from k.
All Science Journal Classification (ASJC) codes
- Applied Mathematics