  We perform the first direct determination of the position of the leading
singularity of the pressure in the complex chemical potential $\mu_B$ plane in
lattice QCD using numerical simulations with 2-stout improved rooted staggered
fermions. This provides a direct determination of the radius of convergence of
the Taylor expansion of the pressure that does not rely on a finite-order
truncation of the expansion. The analyticity issues in the complex $\mu_B$
plane of the grand canonical partition function of QCD with rooted staggered
fermions are solved with a careful redefinition of the fermion determinant that
makes it a polynomial in the fugacity on any finite lattice, without changing
the continuum limit of the observables. By performing a finite volume scaling
study at a single coarse lattice spacing, we show that the limiting singularity
is not on the real line in the thermodynamic limit, thus showing that the
radius of convergence of the Taylor expansion gives a lower bound on the
location of a possible phase transition. In the vicinity of the crossover
temperature at zero chemical potential, the radius of convergence turns out to
be $\mu_B/T \approx 2$ and roughly temperature independent.
