= Solution
The displayed <power series> is the <matrix exponential> $G=e^H$. From $H^3=-\alpha^2H$, its odd powers and nonconstant even powers sum separately:
$$
G=I+\left(1-\frac{\alpha^2}{3!}+\frac{\alpha^4}{5!}-\cdots\right)H+\left(\frac1{2!}-\frac{\alpha^2}{4!}+\frac{\alpha^4}{6!}-\cdots\right)H^2.
$$
The scalar <power series> are $\sin\alpha/\alpha$ and $(1-\cos\alpha)/\alpha^2$, with continuous values $1$ and $1/2$ at zero. Substituting $H\mathbf x=\alpha\mathbf n\times\mathbf x$ and the preceding formula for $H^2\mathbf x$ yields
$$
G\mathbf x=\mathbf x+\sin\alpha\,\mathbf n\times\mathbf x+(1-\cos\alpha)\mathbf n\times(\mathbf n\times\mathbf x).
$$
This is the <Rodrigues rotation formula>: the component parallel to $\mathbf n$ is fixed, while the perpendicular component undergoes a <rotation> through $\alpha$. For $\alpha=0$ the <linear operator> is the <identity operator>.
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