Definition:Harmonic Numbers
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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.
