  We focus on the optimal value for various information-theoretical tasks.
There are several studies for the asymptotic expansion for these optimal values
up to the order $\sqrt{n}$ or $\log n$. However, these expansions have errors
of the order $o(\sqrt{n})$ or $o(\log n)$, which does not goes to zero
asymptotically. To resolve this problem, we derive the asymptotic expansion up
to the constant order for upper and lower bounds of these optimal values. While
the expansions of upper and lower bonds do not match, they clarify the ranges
of these optimal values, whose errors go to zero asymptotically.
