We consider a resummation of the large N expansion of free energy of
pure Chern-Simons theory and the topological string on the resolved
conifold, and find a simple integral formula for the resummed free
energy. Expanding this integral representation around the large
radius point at finite string coupling, we find that it includes not
only the M-theoretic resummation a la Gopakumar and Vafa, but also a
non-perturbative correction in string coupling. Remarkably, the
non-perturbative correction thus obtained is in perfect agreement with
a proposal for membrane instanton corrections given by the refined
topological string in the Nekrasov-Shatashvili limit.
