  We investigate the convergence of the derivative expansion of the exact
renormalization group, by using it to compute the beta function of scalar field
theory. We show that the derivative expansion of the Polchinski flow equation
converges at one loop for certain fast falling smooth cutoffs. The derivative
expansion of the Legendre flow equation trivially converges at one loop, but
also at two loops: slowly with sharp cutoff (as a momentum-scale expansion),
and rapidly in the case of a smooth exponential cutoff. Finally, we show that
the two loop contributions to certain higher derivative operators (not involved
in beta) have divergent momentum-scale expansions for sharp cutoff, but the
smooth exponential cutoff gives convergent derivative expansions for all such
operators with any number of derivatives.
