  A Cluster Expansion Model (CEM), representing a relativistic extension of
Mayer's cluster expansion, is constructed to study baryon number fluctuations
in QCD. The temperature dependent first two coefficients, corresponding to the
partial pressures in the baryon number $B = 1$ and $B = 2$ sectors, are the
only model input, which we fix by recent lattice data at imaginary
baryochemical potential. All other coefficients are constructed in terms of the
first two and required to match the Stefan-Boltzmann limit at $T \to \infty$.
The CEM allows calculations of the baryon number susceptibilities $\chi_k^B$ to
arbitrary order. We obtain excellent agreement with available lattice data for
the baryon fluctuation measures $\chi_2^B$, $\chi_4^B$, $\chi_6^B$ and predict
higher order susceptibilities, that are not yet available from Lattice QCD. The
calculated susceptibilities are then used to extract the radius of convergence
of the Taylor expansion of the pressure. The commonly used ratio test fails due
to the non-trivial asymptotic behavior of the Taylor coefficients. At the same
time, a more elaborate estimator provides finite convergence radii at all
temperatures and in agreement with the singularities of Pad\'e approximants.
The associated singularities lie in the complex $\mu_B/T$-plane and appear
smoothly connected to the Roberge-Weiss transition at high temperatures and
imaginary chemical potential. No evidence for a phase transition at $\mu_B / T
\lesssim \pi$ is found.
