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This note is concerned with series of the forms $\sum f(a^n)$ and $\sumf(n^{-a})$ where f(a) possesses a Mellin transform and $a > 1$ or $a<0$respectively. Integral representations are derived and used to transform theseseries in several ways yielding a selection of interesting integral evaluationsinvolving Riemann's function $\zeta(s)$, limits and series representationscontaining hyperpowers. A number of examples of such sums are provided, each ofwhich is investigated for possible new structure. In one case, we obtain ageneralization of Riemann's classic relationship among the Zeta, Gamma andJacobi Theta functions.

On the Use of the Mellin Transform to Generate Families of Power, Hyperpower, Lambert and Dirichlet Type Series and Some Consequences