  By reduction along the time direction, black holes in 4 dimensions yield
instantons in 3 dimensions. Each of these instantons contributes individually
at order $\exp(-|Q|/g_s)$ to certain protected couplings in the
three-dimensional effective action, but the number of distinct instantons is
expected to be equal (or comparable) to the number of black hole micro-states,
i.e. of order $\exp(Q^2)$. The same phenomenon also occurs for certain
protected couplings in four dimensions, such as the hypermultiplet metric in
type II string theories compactified on a Calabi-Yau threefold. In either case,
the D-instanton series is therefore asymptotic, much like the perturbative
expansion in any quantum field theory. By using a Borel-type resummation
method, adapted to the Gaussian growth of the D-instanton series, we find that
the total D-instanton sum has an inherent ambiguity of order $\exp(-1/g_s^2)$.
We further suggest that this ambiguity can be lifted by including Kaluza-Klein
monopole or NS5-brane instantons.
