  It has recently been proposed that a class of supersymmetric
higher-derivative interactions in N=2 supergravity may encapsulate an infinite
number of finite size corrections to the microscopic entropy of certain
supersymmetric black holes. If this proposal is correct, it allows one to probe
the string theory description of black-hole micro-states to far greater
accuracy than has been possible before. We test this proposal for ``small''
black holes whose microscopic degeneracies can be computed exactly by counting
the corresponding perturbative BPS states. We also study the ``black hole
partition sum'' using general properties of of BPS degeneracies. This
complements and extends our earlier work in hep-th/0502157
