Consider a real function \(f\) that can be approximated with the asymptotic expansion

\[f_{n}(x)=\pi x+\frac{1}{2} n^{-1 / 2} \pi^{2} x^{1 / 2}-\frac{1}{3} n^{-1} \pi^{3} x^{1 / 4}+O\left(n^{-3 / 2}ight),\]

as \(n ightarrow \infty\), uniformly in \(x\), where \(x\) is assumed to be positive. Use the first method demonstrated in Section 1.6 to find an asymptotic expansion with error \(O\left(n^{-3 / 2}ight)\) as \(n ightarrow \infty\) for \(x_{a}\) where \(f\left(x_{a}ight)=a+O\left(n^{-3 / 2}ight)\) as \(n ightarrow \infty\).