  Within about a year (1916-1917) Chapman and Enskog independently proposed an
important expansion for solving the Boltzmann equation. However, the expansion
is divergent or indeterminant in the case of relaxation time $\tau \geq 1$.
Even since this divergence problem has puzzled this subject for a century. By
using a modified M\"obius series inversion formula, this paper proposes a
modified Chapman-Enskog expansion with a variable upper limit of the summation.
The new expansion can give not only a convergent summation but also provide the
best-so-far explanation on some unbelievable scenarios occurred in previous
practice.
