  We study the large-order asymptotic behavior of the perturbation series for
short-distance coefficients in the NRQCD factorization formulas for the decays
J/psi --> e^+e^- and eta_c --> gamma gamma. The short-distance coefficients of
the leading matrix elements are calculated to all orders in the large-N_f
limit. We find that there is a universal Borel resummable renormalon associated
with the cancellation of the Coulomb singularity in the short-distance
coefficients. We verify that the ambiguities in the short-distance coefficients
from the first infrared renormalon are canceled by ambiguities in the
nonperturbative NRQCD matrix elements that contribute through relative order
v^2. Our results are used to estimate the coefficients of higher order
radiative corrections in the decay rates for J/psi --> e^+ e^-, eta_c --> gamma
gamma, and J/\psi --> gamma gamma gamma.
