Harmonic numbers

The set $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$ of harmonic numbers is a time scale.

$\mathbb{T}=\mathbb{H}$

Generic element $t\in \mathbb{T}$:	For some $n \in \mathbb{Z}^+, t =\displaystyle\sum_{k=0}^n \dfrac{1}{k}$
Jump operator:	$\begin{array}{ll} \sigma(t) &= \sigma \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right) \\ &= \displaystyle\sum_{k=1}^{n+1} \dfrac{1}{k} \\ &= t + \dfrac{1}{n+1} \\ \end{array}$
Graininess operator:	$\begin{array}{ll} \mu(t) &= \mu \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right) \\ &= t + \dfrac{1}{n+1} - t \\ &= \dfrac{1}{n+1} \\ \end{array}$
$\Delta$-derivative:	$\begin{array}{ll} f^{\Delta}(t) &= f^{\Delta} \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k} \right) \\ &= \dfrac{f \left( \displaystyle\sum_{k=1}^{n+1} \dfrac{1}{k} \right) - f \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k} \right)}{\mu \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k} \right)} \\ &= (n+1) \left[ f \left( \displaystyle\sum_{k=1}^{n+1} \dfrac{1}{k} \right) - f \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k} \right) \right] \end{array}$
$\Delta$-integral:	$\begin{array}{ll} \displaystyle\int_s^t f(\tau) \Delta \tau &= \displaystyle\int_{\sum_{k=1}^m \frac{1}{k}}^{\sum_{k=1}^n \frac{1}{k}} f(\tau) \Delta \tau \\ &= \displaystyle\sum_{k=m}^{n-1}\dfrac{1}{k+1} f \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k} \right) \end{array}$
Exponential function:	$\begin{array}{ll} e_p(t,s) &= e_p \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k}, \displaystyle\sum_{k=1}^m \dfrac{1}{k} \right) \\ &= \exp \left( \displaystyle\int_{ \sum^m \frac{1}{k}}^{\sum^n \frac{1}{k}} \dfrac{1}{\mu(\tau)} \log(1 + \mu(\tau) p(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=m}^{n-1} \log \left[1 + \mu \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) \right] \right) \\ &= \exp \left( \displaystyle\sum_{k=m}^{n-1} \log \left[1 + \dfrac{1}{k+1} p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) \right] \right) \\ &= \displaystyle\prod_{k=m}^{n-1} 1 + \dfrac{1}{k+1} p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) \\ \end{array}$

Examples of time scales
$\Huge\mathbb{R}$ Real numbers	$\Huge\mathbb{Z}$ Integers	$\Huge{h\mathbb{Z}}$ Multiples of integers	$\Huge\mathbb{Z}^2$ Square integers	$\Huge\mathbb{H}$ Harmonic numbers	$\Huge\mathbb{T}_{\mathrm{iso}}$ Isolated points
$\Huge\sqrt[n]{\mathbb{N}_0}$ nth root numbers	$\Huge\mathbb{P}_{a,b}$ Evenly spaced intervals	$\huge\overline{q^{\mathbb{Z}}}$ Quantum, $q>1$	$\huge\overline{q^{\mathbb{Z}}}$ Quantum, $q<1$	$\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$ Closure of unit fractions	$\Huge\mathcal{C}$ Cantor set