  Comparing phase plots of truncated series solutions of Kepler's equation by
Lagrange's power series with those by Bessel's Kapteyn series strongly suggest
that a Jentzsch-type theorem holds true not only for the former but also for
the latter series: each point of the boundary of the domain of convergence in
the complex plane is a cluster point of zeros of sections of the series. We
prove this result by studying properties of the growth function of a sequence
of entire functions. For series, this growth function is computable in terms of
the convergence abscissa of an associated general Dirichlet series. The proof
then extends, besides including Jentzsch's classical result for power series,
to general Dirichlet series, to Kapteyn, and to Neumann series of Bessel
functions. Moreover, sections of Kapteyn and Neumann series generally exhibit
zeros close to the real axis which can be explained, including their asymptotic
linear density, by the theory of the distribution of zeros of entire functions.
