Harmonic Numbers: Insights, Approximations and Applications

The applications of spreadsheets to mathematics allow learners to analyze mathematical ideas in ways never before possible. In this paper harmonic numbers are defined, calculated, approximated, and applied. The concepts and techniques of recursion, approximation and combinatorial methods are illustrated. This paper also shows some properties of the harmonic series including divergence and its part in the determination of harmonic numbers. Analyzing the ideas of harmonic numbers presents to learners that one concept has many connections to others. Mathematical concepts and relationships between harmonic numbers, harmonic series, the Euler-Mascheroni constant and the harmonic mean are explored. Finding expressions in closed-form is not always possible and technology provides the capability to approximate the harmonic numbers.\nExploration and experimentation are discussed. With spreadsheets, complex calculations can be simplified, intuition can be developed and applications can be studied.