Opposite point charges have no linear restoring force at their midpoint
= Opposite point charges have no linear restoring force at their midpoint
{title2=$F(f)=2k_eqq_3L^{-2}(1+3f^2+O(f^4))$}
Place $+q$ at $-L$ and $-q$ at $+L$. For a nonzero <point charge> $q_3$ between them, the signed force at $x=fL$ is $2k_eqq_3(1+f^2)/[L^2(1-f^2)^2]$. It is nonzero at $f=0$ and even in $f$, so its first nonconstant term is quadratic. The midpoint is not an <equilibrium> and cannot be treated as a harmonic restoring centre.