  We review the closed-forms of the partial Fourier sums associated with
$HP_k(n)$ and create an asymptotic expression for $HP(n)$ as a way to obtain
formulae for the full Fourier series (if $b$ is such that $|b|<1$, we get a
surprising pattern, $HP(n) \sim H(n)-\sum_{k\ge 2}(-1)^k\zeta(k)b^{k-1}$).
Finally, we use the found Fourier series formulae to obtain the values of the
Lerch transcendent function, $\Phi(e^m,k,b)$, and by extension the
polylogarithm, $\mathrm{Li}_{k}(e^{m})$, at the positive integers $k$.
