= Solution
At $v_*=V-\sigma_b$, the beam's positive slope is $n_b e^{-1/2}/(\sqrt{2\pi}\sigma_b^2)$, while the background slope has magnitude $n_pv_*e^{-v_*^2/(2\sigma_p^2)}/(\sqrt{2\pi}\sigma_p^3)$. Thus a nonnegative total slope at this velocity requires
$$
\frac{n_b}{n_p}\ge\sqrt e\,\frac{v_*\sigma_b^2}{\sigma_p^3}e^{-v_*^2/(2\sigma_p^2)}.
$$
Approximating the background slope by its value at $V$ yields
$$
\boxed{\frac{n_b}{n_p}>C\sqrt{\frac{m_e}{k_{\rm B}T_p}}\frac{VT_b}{T_p}\exp\left(-\frac{m_eV^2}{2k_{\rm B}T_p}\right),\qquad C=\sqrt e.}
$$
Equality is the local marginal-slope condition; strict inequality gives positive resonant drive in this approximation. The numerical $C$ is of order unity and can change when the resonant velocity is optimized or the full wave dispersion is retained.
Replacing $v_*$ by $V$ in the exponential is controlled only if $V\sigma_b/\sigma_p^2\ll1$, in addition to $\sigma_b\ll\sigma_p$. The two width inequalities alone do not guarantee that extra condition far out on a Maxwellian tail. The preceding unapproximated slope inequality remains available when it is not satisfied.
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