Notice near $y=1$, the function

$$(1-y^{-2})^{-1/n}=(y-1)^{-1/n}\times\text{Taylor series in $y-1$ with constant term $2^{-1/n}$}.$$

Upon integration,

$$F(x)=(x-1)^{(n-1)/n}\times\text{Taylor series in $x-1$ with constant term $2^{-1/n}\frac{n}{n-1}$}.$$

Raising (18) to the power $n/(n-1)$, we see

$$w_{0}^{-\frac{n+2}{n-1}}(s-1)^{n/(n-1)}=\text{Taylor series in $\frac{\mathfrak{w}}{w_{0}}-1$ with first coefficient $\frac{n}{n-1}$}.$$

Inverting the function, $\frac{\mathfrak{w}}{w_{0}}-1$ is a power series of $(s-1)^{n/(n-1)}$ near $s=1$. To leading order,

$$\frac{\mathfrak{w}}{w_{0}}-1=\frac{n-1}{n}w_{0}^{-\frac{n+2}{n-1}}(s-1)^{\frac{n}{n-1}}\left(1+O((s-1)^{\frac{n}{2n-1}})\right),$$

$$\frac{d\mathfrak{w}}{ds}\approx w_{0}^{-\frac{3}{n-1}}(s-1)^{1/(n-1)}=b_{1}(s-1)^{1/(n-1)},$$

which agrees with the initial condition (16). ∎