In Kubo's Statistical Mechanics, his Ex 4.1 has a line that goes from a relation between $\mu$ and $\mu_0$, $$ \mu = \mu_0 \left( 1 - \frac{\pi^2}{12}\left(\frac{kT}{\mu}\right)^2 + \frac{\pi^4}{720}\left(\frac{kT}{\mu}\right)^4 +\dots \right) $$
to a function $\mu(\mu_0)$ $$ \mu = \mu_0 \left( 1 - \frac{\pi^2}{12}\left(\frac{kT}{\mu_0}\right)^2 - \frac{\pi^4}{80}\left(\frac{kT}{\mu_0}\right)^4 +\dots \right) $$
Did Kubo assume a power series form $\frac{kT}{\mu}=\sum c_n \left(\frac{kT}{\mu_0}\right)^n$, and somehow use this?
A helpful answer can be found by wiki "Lagrange inversion theorem" where a general method of computing the Taylor expansion of the inverse function:
https://en.wikipedia.org/wiki/Lagrange_inversion_theorem