Some series representing the zeta function for $\mathrm{\Re }s>1$

We represent the Riemann zeta function in the half-plane $\mathrm{\Re }s>1$ via series whose terms admit geometrically decreasing bounds. Due to an underlying recurrence relation, which is used to compute coefficients entering into the terms, the cost is probably at least quadratic in the number of terms. And the number of terms needed to reach a given fixed-point precision grows linearly with the imaginary part, so, presumably, the usefulness is limited to small imaginary parts (up to the hundreds perhaps). The method is a development of tools introduced by the author for the evaluation of harmonic series with restricted digits in a given radix.