We provide representations of Euler's constant $γ=0.577...$ 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-Erdélyi asymptotic expansion of the ratio of two Gamma functions, is established for that purpose.