  We derive the large order behavior of the perturbative expansion for the
continuous model of tethered self-avoiding membranes. It is controlled by a
classical configuration for an effective potential in bulk space, which is the
analog of the Lipatov instanton, solution of a highly non-local equation. The
n-th order is shown to have factorial growth as (-cst)^n (n!)^(1-epsilon/D),
where D is the `internal' dimension of the membrane and epsilon the engineering
dimension of the coupling constant for self-avoidance. The instanton is
calculated within a variational approximation, which is shown to become exact
in the limit of large dimension d of bulk space. This is the starting point of
a systematic 1/d expansion. As a consequence, the epsilon-expansion of
self-avoiding membranes has a factorial growth, like the epsilon-expansion of
polymers and standard critical phenomena, suggesting Borel summability.
Consequences for the applicability of the 2-loop calculations are examined.
