Using Stirling's formula as before, we can obtain an approximation of
for large *l* values. Expanding the first and third terms in (66) as
Taylor series reveals that *f* can be approximated by
.
Although it
is not obvious that *f* should have this form, the negative sign for the correction term does
make sense. Even though the eigenvalues become more evenly distributed as *n* increases,
there are a large number of them at the endpoint
that are always excluded
from the interval *I*_{n}. The correction term may reflect the effect of this
exclusion.

**Remark.** The results derived in this paper are for half-open intervals, rather
than for more standard open or closed intervals. This was done mainly to simplify notation as
much as possible. In fact, the limit for an open interval of the form