  The main purpose of this paper is to derive a new perturbation theory (PT)
that has converging series. Such series arise in the nonlocal scalar quantum
field theory (QFT) with fractional power potential. We construct PT for the
generating functional (GF) of complete Green functions (including disconnected
parts of functions) $\mathcal{Z}\left[j\right]$ as well as for GF of connected
Green functions $\mathcal{G}\left[j\right]=\ln \mathcal{Z}\left[j\right]$ in
powers of coupling constant $g$. It has infrared (IR)-finite terms. We prove
that the obtained series, which has the form of a grand canonical partition
function (GCPF), is dominated by a convergent series, in other words, has
majorant, which allows to expand beyond the weak coupling $g$ limit. Vacuum
energy density in second order in $g$ is calculated and researched for
different types of Gaussian part $S_{0}[\phi]$ of the action $S[\phi]$. Further
in the paper, using the polynomial expansion, the general calculable series for
$\mathcal{G}\left[j\right]$ is derived. We provide, compare and research
simplifications in cases of second-degree polynomial and hard-sphere gas (HSG)
approximations. The developed formalism allows us to research the physical
properties of the considering system across the entire range of coupling
constant $g$, in particular, the vacuum energy density.
