= Solution
Let $\kappa=kV_e/\omega_{\mathrm pe}$ and $s=\omega/(kV_e)$. The long-<wavelength> electron branch has $|s|\gg1$. The convergent <power series>
$$
\log\frac{s-1}{s+1}=-2\left(\frac1s+\frac1{3s^3}+\frac1{5s^5}+\cdots\right)
$$
turns the <degenerate-plasma dielectric response> into
$$
0=1-\frac{\omega_{\mathrm pe}^2}{\omega^2}
-\frac{3\omega_{\mathrm pe}^2 k^2V_e^2}{5\omega^4}
+O\left(\frac{\omega_{\mathrm pe}^2 k^4V_e^4}{\omega^6}\right).
$$
Substitute $\omega^2=\omega_{\mathrm pe}^2+a k^2V_e^2+O(k^4V_e^4/\omega_{\mathrm pe}^2)$. The coefficient of $k^2$ is $a-3/5$, so the <degenerate-plasma Langmuir dispersion> is
$$
\boxed{\omega^2=\omega_{\mathrm pe}^2+\frac35 k^2V_e^2+O\left(\frac{k^4V_e^4}{\omega_{\mathrm pe}^2}\right),\qquad
\omega_\pm=\pm\omega_{\mathrm pe}\left[1+\frac3{10}\kappa^2+O(\kappa^4)\right].}
$$
The <Langmuir wave> therefore has a finite <frequency> at zero <wavenumber>, with the first dispersive correction set by the <Fermi velocity>. Indeed the occupied <Fermi sphere> has $\langle v_\parallel^2\rangle=V_e^2/5$, so the coefficient $3\langle v_\parallel^2\rangle=3V_e^2/5$ agrees with the high-<phase velocity> <velocity moment> expansion.
\b[Screening-length convention.] The source defines $\lambda_{\mathrm De}^{\rm exam}=\sqrt3 V_e/\omega_{\mathrm pe}$, so its long-<wavelength> condition is equivalent to $\kappa\ll1$. We retain that definition when expressing answers in its symbols. The actual electron <Thomas-Fermi screening length> obtained from the static response is instead
$$
\epsilon_e(0,k)=1+\frac{3\omega_{\mathrm pe}^2}{k^2V_e^2}
=1+\frac1{k^2\lambda_{\rm TF}^2},\qquad
\boxed{\lambda_{\rm TF}=\frac{V_e}{\sqrt3\omega_{\mathrm pe}}=\frac{\lambda_{\mathrm De}^{\rm exam}}3.}
$$
For a concrete comparison, at $k=\omega_{\mathrm pe}/V_e$, the static <dielectric function> is $4$; using the printed length as the physical <Thomas-Fermi screening length> would give $1+1/3=4/3$. Calling the printed length the <Debye length> thus differs from the physical degenerate-electron screening convention by a factor of three. It does not affect the boxed dimensional <dispersion relation>.
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