Absolute convergence of a weight-k real-analytic Eisenstein series
= Absolute convergence of a weight-k real-analytic Eisenstein series
The absolute value of the summand indexed by a primitive bottom row $(c,d)$ is
$$
y^{\operatorname{Re}s}|c\tau+d|^{-2\operatorname{Re}s-k}.
$$
The exponent exceeds two, so comparison with the lattice sum over $(c,d)\in\mathbb Z^2\setminus\{0\}$ proves absolute and locally uniform convergence.