= Solution
For a <plane wave> <wavefunction> $\psi_k(x)\propto e^{ikx}$, impose the cyclic or <periodic boundary condition> $\psi_k(x+L)=\psi_k(x)$. It requires
$$
e^{ikL}=1,\qquad k_n=\frac{2\pi n}{L},\quad n\in\mathbb Z.
$$
Adjacent allowed <wavenumbers> on the full signed $k$ axis are spaced by $\Delta k=2\pi/L$. Ignoring <spin>, the <density of states> per unit $k$ is consequently
$$
\boxed{\rho_k=\frac1{\Delta k}=\frac{L}{2\pi}.}
$$
Both positive and negative $k$ are included in this density. It is not yet the density per unit magnitude $|k|$, which has an extra factor of two away from zero. In $d$ independent dimensions the same <periodic boundary conditions> give one spatial state per $k$-space cell of measure $(2\pi/L)^d$.
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