Some geometric series for Euler's constant

We provide representations of Euler’s constant $\gamma =\mathrm{0.577...}$ as series which converge geometrically fast (but use a certain sequence whose computation induces a quadratic cost). The asymptotic oscillations of these coefficients are determined to all orders. A result of independent interest, about sufficient conditions for the validity, in the case of unbounded parameters, for the Tricomi-Erd'elyi asymptotic expansion of the ratio of two Gamma functions, is established for that purpose.