  We develop a new formulation of Stein's method to obtain computable upper
bounds on the total variation distance between the geometric distribution and a
distribution of interest. Our framework reduces the problem to the construction
of a coupling between the original distribution and the "discrete equilibrium"
distribution from renewal theory. We illustrate the approach in four
non-trivial examples: the geometric sum of independent, non-negative,
integer-valued random variables having common mean, the generation size of the
critical Galton-Watson process conditioned on non-extinction, the in-degree of
a randomly chosen node in the uniform attachment random graph model and the
total degree of both a fixed and randomly chosen node in the preferential
attachment random graph model.
