  Let $v_n$ be the maximum expected length of an increasing subsequence, which
can be selected by an online nonanticipating policy from a random sample of
size $n$. Refining known estimates, we obtain an asymptotic expansion of $v_n$
up to a $O(1)$ term. The method we use is based on detailed analysis of the
dynamic programming equation, and is also applicable to the online selection
problem with observations occurring at times of a Poisson process.
