The Taylor expansion of the CDF of the exponential function centered at $u_n$ -- with $u_n$ being an arbitrarily large value defined as $F(u_n)=1-\frac 1 n$ -- is
$$F(x) = 1 - \frac 1 n e^{\alpha_n(x-u_n)}$$
and
$$\alpha_n:=nf(u_n)$$
according to page 21 Statistical Theory of Extreme Values and Some Practical Applications
or so the following seems to indicate:
The asymptotic distribution of the largest value for the exponential type may be derived in the following way: The mode $\bar x_n$ of the largest value is obtained from equation $( 3.2)$ as the solution of
$$\frac{n-1}{F(x)}f(x) =-\frac{f'(x)}{f(x)}$$
If $x$ becomes very large, the density of probability $f (x )$ becomes very small , and the same holds for the probability $1 -F(x)$ of a value surpassing $x.$ If the variate is unlimited, the derivative $f ' > ( x )$ also converges toward zero . If it is legitimate from a certain value $x$ onward to apply l'Hôpital's rule and obtain the relation
$$\frac{f(x)}{1-F(x)} \sim-\frac{f'(x)}{f(x)}$$
If the same rule may be applied to the next derivatives , the Taylor expansion of the probability function in the neighborhood of $u_n$ converges toward
$$F(x) = 1 - \frac 1 n e^{\alpha_n(x-u_n)}$$
I don't see how this equation is reached.
I think I got a work-around this alpha parameter thanks to Gumbel's original paper in French:
The base experiment is to draw a large number $(N)$ values from a distribution $F(x)$ (symbolized as $W(x)$ in the original, and consider the highest $m$ realizations, ordered as $x_1, x_2,\dots,x_m,\dots,x_N$ in decreasing order.
Let $\tilde u_m$ be an observed value such that there are $m$ values as high or higher (Soit $\tilde u_m$ une grandeur observée, telle qu’il y enait $m$ autres supérieures ou égales (p.123)). The probability of obtaining a value as great or greater than $\tilde u_m$ is $1-F(\tilde u_m).$ Now (on the same page),
$$\small 1-F(\tilde u_m)=1-\frac m N$$
applying L'Hôpital's rule to $$\small\frac{f(x)}{1-F(x)} \sim-\frac{f'(x)}{f(x)}\tag {*}$$ (p. 118, and also p. 21 of Gumbel's book in English) This is explained here.
The Taylor expansion of the original distribution around $\tilde u_m$ (p. 132) is
$$\begin{align} F(x) &= F(\tilde u_m)+ \frac{x-\tilde u_m}{1!}f(\tilde u_m)+ \frac{(x-\tilde u_m)^2}{2!}f'(\tilde u_m)+\cdots\\[2ex] &\small= 1 - \frac m N + \frac m N \frac{x-\tilde u_m}{1!} \frac N m f(\tilde u_m) +-\frac m N \frac{(x-\tilde u_m)^2}{2!} \frac{N^2}{m^2} \big [f(\tilde u_m) \big ]^2 +\cdots \tag 1 \end{align}$$
the last bit from $(*)$:$ f'(x)= \frac{[f(x)]^2}{-(1-F(x))}.$
This corresponds to the first three terms in the expansion of
$$1-\frac m N \exp\left (-(x-\tilde u_m)\frac N m f(\tilde u_m)\right) \tag 2$$
Equating $(1)$ and $(2),$ and introducing $y_m=\frac N m f(\tilde u_m)(x-\tilde u_m),$ and considering that (p. 19 English version)
$$\Phi_N(\tilde u_m)=F^N(\tilde u_m)$$
which stands for the probability distribution that the largest value of a sample of $N$ independent observations from the probability $F$ - i.e. the probability that the largest value is less than $\tilde u_m,$
$$\small\Phi_N(\tilde u_m) = \left(1-\frac m N\exp\left (-y_m\right)\right)^N =\left(1-\frac {m\,e^{-y_m}} N\right)^N$$
which for large samples $N\to \infty$ is equivalent to the continuous compounding formula resulting in the final Gumbel CDF with the double exponential:
$$\Phi_N(x)= e^{-m \,e^{-y}} .$$