= Solution
The <Gumbel distribution> for maxima has <distribution function> $G(x)=\exp(-e^{-x})$ on the whole real line. For $n\geq2$ set
$$
\boxed{\beta_n=(\log n)^2,\qquad \alpha_n=2\log n.}
$$
Both the centering and scale are elementary, and the scale is positive. For any fixed real $x$, the normalized threshold $\beta_n+\alpha_nx$ is positive for all sufficiently large $n$. Writing $L=\log n$, a <Taylor expansion> gives
$$
\sqrt{\beta_n+\alpha_nx}
=L\sqrt{1+\frac{2x}{L}}
=L+x+O(L^{-1}).
$$
Consequently,
$$
n\overline F(\beta_n+\alpha_nx)
=n\exp\!\left(-\sqrt{\beta_n+\alpha_nx}\right)
\longrightarrow e^{-x}.
$$
The logarithm calculation in part (i) now gives
$$
\boxed{F(\alpha_nx+\beta_n)^n\longrightarrow\exp(-e^{-x})\quad(x\in\mathbb R).}
$$
Thus $F$ belongs to the <maximum domain of attraction> of the standard <Gumbel distribution>. The scale also has the <Gumbel auxiliary function> interpretation: the reciprocal <hazard function> is $2\sqrt t$, whose <derivative> $1/\sqrt t$ tends to zero at the infinite right endpoint. This agrees with the <Von Mises conditions for extreme values>, and at $t=\beta_n$ gives exactly $\alpha_n=2\log n$.
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