Partial sums of the hyperharmonic series

TL;DR

This paper proves that, similar to the harmonic series, no two partial sums of the hyperharmonic series are equal, extending a classical result to a broader class of series.

Contribution

It generalizes Erdös and Niven's result from the harmonic series to the hyperharmonic series, establishing the uniqueness of partial sums.

Findings

No two partial sums of the hyperharmonic series are equal.

The result extends classical harmonic series properties to hyperharmonic series.

Provides a new understanding of the structure of hyperharmonic series.

Abstract

In 1946, Erd\"os and Niven proved that no two partial sums of the harmonic series are equal. In this paper, we extend this result by demonstrating that no two partial sums of the hyperharmonic series are equal.