Proof of Divergence of the Perturbative Series for a Large Collection of Scalar Field Models in Euclidean Lattice Quantum Field Theory

We present a simple but rigorous mathematical proof, based on well-known
theorems about power series in the theory of analytic functions, that
the perturbative series of a large number of scalar field models in
lattice quantum field theory, with interaction terms that are given by
$\lambda\varphi^{2p}$ for $p\geq 2$, are divergent at all points except
their points of reference $\lambda=0$. This is true not only in the
continuum limit, but on every individual finite lattice as well, for all
spacetime dimensions $d\geq 3$.