  It is a classical fact that domains of convergence of power series of several
complex variables are characterized as logarithmically convex complete
Reinhardt domains; let $D \subsetneq \mathbb{C}^N$ be such a domain. We show
that a necessary as well as sufficient condition for a power series $g$ to have
$D$ as its domain of convergence is that it admits a certain decomposition into
elementary power series; specifically, $g$ can be expressed as a sum of a
sequence of power series $g_n$ with the property that each of the logarithmic
images $G_n$ of their domains of convergence are half-spaces, all containing
the logarithmic image $G$ of $D$ and such that the largest open subset of
$\mathbb{C}^N$ on which all the $g_n$'s and $g$ converge absolutely is $D$. In
short, every power series admits a decomposition into elementary power series.
The proof of this leads to a new way of arriving at a constructive proof of the
aforementioned classical fact. This proof inturn leads to another decomposition
result in which the $G_n$'s are now wedges formed by intersections of pairs of
{\it supporting} half-spaces of $G$. Along the way, we also show that in each
fiber of the restriction of the absolute map to the boundary of the domain of
convergence of $g$, there exists a singular point of $g$.
