Definition:Harmonic Numbers

Definition
The harmonic numbers are denoted $H_n$ and are defined for positive integers $n$:
 * $\displaystyle \forall n \in \Z, n \ge 0: H_n = \sum_{k=1}^n \frac 1 k$

From the definition of vacuous summation it is clear that $H_0 = 0$.

From Sum of Reciprocals is Divergent it is clear that $H_n$ is unbounded above.

Generalized Harmonic Numbers
When $r \in \R: r > 1$, we define the following:
 * $\displaystyle H_n^{(r)} = \sum_{k=1}^n \frac 1 {k^r}$

and we note that $\displaystyle H_\infty^{(r)} = \lim_{n \to \infty} H_n^{(r)}$ is in fact the Riemann zeta function.

From P-Series Converge Absolutely, we have that $H_n^{(r)}$ is bounded for all $r > 1$.

Notation
There is no standard notation for this series.

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