Definition:Harmonic Numbers/General Definition

This page is about General Harmonic Numbers. For other uses, see Harmonic.

Definition

Let $r \in \R_{>0}$.

For $n \in \N_{> 0}$ the Harmonic numbers order $r$ are defined as follows:

$\ds H_n^{\paren r} = \sum_{k \mathop = 1}^n \frac 1 {k^r}$

Notation

There is no standard notation for this series.

The notation given here is as advocated by Donald E. Knuth.

Also see

• Definition:Riemann Zeta Function: $\ds H_\infty^{\paren r} = \lim_{n \mathop \to \infty} H_n^{\paren r}$

• Definition:P-Series

• Results about general harmonic numbers can be found here.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: $(4)$