  We design algorithms for computing approximately revenue-maximizing {\em
sequential posted-pricing mechanisms (SPM)} in $K$-unit auctions, in a standard
Bayesian model. A seller has $K$ copies of an item to sell, and there are $n$
buyers, each interested in only one copy, who have some value for the item. The
seller must post a price for each buyer, the buyers arrive in a sequence
enforced by the seller, and a buyer buys the item if its value exceeds the
price posted to it. The seller does not know the values of the buyers, but have
Bayesian information about them. An SPM specifies the ordering of buyers and
the posted prices, and may be {\em adaptive} or {\em non-adaptive} in its
behavior.
  The goal is to design SPM in polynomial time to maximize expected revenue. We
compare against the expected revenue of optimal SPM, and provide a polynomial
time approximation scheme (PTAS) for both non-adaptive and adaptive SPMs. This
is achieved by two algorithms: an efficient algorithm that gives a
$(1-\frac{1}{\sqrt{2\pi K}})$-approximation (and hence a PTAS for sufficiently
large $K$), and another that is a PTAS for constant $K$. The first algorithm
yields a non-adaptive SPM that yields its approximation guarantees against an
optimal adaptive SPM -- this implies that the {\em adaptivity gap} in SPMs
vanishes as $K$ becomes larger.
