  In this note we present some results on the convergence of Nekrasov partition
functions as power series in the instanton counting parameter. We focus on
$U(N)$ ${\mathcal N}=2$ gauge theories in four dimensions with matter in the
adjoint and in the fundamental representations of the gauge group respectively
and find rigorous lower bounds for the convergence radius in the two cases: if
the theory is {\it conformal}, then the series has at least a {\it finite}
radius of convergence, while if it is {\it asymptotically free} it has {\it
infinite} radius of convergence. Via AGT correspondence, this implies that the
related irregular conformal blocks of $W_N$ algebrae admit a power expansion in
the modulus converging in the whole plane. By specifying to the $SU(2)$ case,
we apply our results to analyse the convergence properties of the corresponding
Painlev\'e $\tau$-functions.
