Definition:Harmonic Numbers

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

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:
 * $$H_n^{(r)} = \sum_{k=1}^n \frac 1 {k^r}$$

and we note that $$H_\infty^{(r)} = \lim_{n \to \infty} H_n^{(r)}$$ 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.