Some series representing the eta function for $\mathrm{\Re }s>0$

We represent the Euler alternating series (sometimes called the “Dirichlet eta function”), and generally $(b^s-b)\zeta (s)\mathrm{/}{b}^s$ for $b>1$ an integer, in the half-plane $\mathrm{\Re }s>0$, via series dominated by geometric series, with arbitrarily small convergence ratio (up to the prize of a longer first approximation). Due to the underlying recurrence, the cost for each new term is at first sight linearly increasing, so the cost appears to be quadratic in the number of terms kept. And the number of terms needed to achieve a given target precision increases linearly with the imaginary part of $s$.