Monotone and fast computation of Euler’s constant

We construct sequences of finite sums [Formula: see text] and [Formula: see text] converging increasingly and decreasingly, respectively, to the Euler-Mascheroni constant γ at the geometric rate 1/2. Such sequences are easy to compute and satisfy complete monotonicity-type properties. As a consequence, we obtain an infinite product representation for [Formula: see text] converging in a monotone and fast way at the same time. We use a probabilistic approach based on a differentiation formula for the gamma process.