The virial expansion for a two-dimensional system of hard disks gives the following series when expressed in terms of the two-dimensional packing fraction \(\eta=\pi n D^{2} / 4\) :

\[
\begin{aligned}
\frac{P}{n k T}= & 1+2 \eta+3.128018 \eta^{2}+4.257854 \eta^{3}+5.33689664 \eta^{4}+6.363026 \eta^{5} \\
& +7.352080 \eta^{6}+8.318668 \eta^{7}+9.27236 \eta^{8}+10.2161 \eta^{9}+\cdots
\end{aligned}
\]

see Malijevsky and Kolafa (2008). Propose some simple analytical functions \(f(\eta)\) that closely approximate this series.