The master’s thesis discusses harmonic numbers These prove to be very useful in the field of number theory, analysis and probability theory, but they are also found in the analysis of computer algorithms and in various mathematical puzzles.
Before discussing harmonic numbers, we present a complete selection of basic theorems and definitions of sequences, and series that will help us understand the core of the master’s thesis. In this part, we define a harmonic type and prove its divergence. In the following, we focus on harmonic numbers and present some of their properties. In this set, we prove Bertrand’s postulate and use it to show that no harmonic number Hn, n ≥ 2 is an integer. We also show that all harmonic numbers except H1, H2 in H6, have an infinite period. Harmonic numbers are then connected to the natural logarithm. Thereby, we introduce the Euler-Mascheroni constant γ and show two principles by which Euler calculated its decimal approximations. In the following, we define three different representations of har-monic numbers, namely integral representation, representation with binomial symbols, and combinatorial interpretation. We also derive the generic function of harmonic numbers and introduce zeta function and generalized harmonic numbers. We prove that zeta function converges for m > 1 and that ζ(2) = π2.
In the fourth chapter of the thesis, by using combinatorial proofs, we prove the combinatorial interpretation of harmonic numbers and a rounded selection of combinatorial identities containing harmonic numbers. Combinatorial identities are proved mainly by the double counting method.
In the last chapter, we present further examples of the use of harmonic numbers, e. g., the problem of a picture collector, the jeep problem, the problem of hundred prisoners, and the block-stacking problem. We also simulate the given problems by using Python programming language.
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