Definition:Harmonic Numbers/General Definition

Definition
Let $r \in \R_{> 1}$.

For $n \in \N_{> 0}$ the Harmonic numbers order $r$ are defined as follows:
 * $\displaystyle H_n^{\left({r}\right)} = \sum_{k \mathop = 1}^n \frac 1 {k^r}$

Also see

 * P-Series Converges Absolutely, whence $H_n^{\left({r}\right)}$ is bounded for all $r > 1$.


 * Note that $\displaystyle H_\infty^{\left({r}\right)} = \lim_{n \to \infty} H_n^{\left({r}\right)}$ is the Riemann zeta function.

Notation
There is no standard notation for this series.

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