= Solution
The exact energies and mixing coefficients contain
$$
\sqrt{\frac{\Delta^2}{4}+\lambda^2}.
$$
As a function of complex $\lambda$, its nearest <branch points> to the origin occur when its argument vanishes:
$$
\lambda=\pm\frac{i\Delta}{2}.
$$
Their distance from the expansion point $\lambda=0$ determines the <radius of convergence>. Therefore
$$
\boxed{R=\frac{\Delta}{2}=\frac{E_+-E_-}{2}.}
$$
The real-axis perturbation series converges for $|\lambda|<(E_+-E_-)/2$.
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