Multipole expansion of three collinear charges

For charges $-Q,NQ,-MQ$ at $z=d,0,-d$, respectively,The monopole vanishes when $N=M+1$, the dipole vanishes when $M=1$, and both vanish when $(N,M)=(2,1)$.

Subject to $N=M+1$, keeping $Qd$ fixed as $d\to 0$ gives a finite point-dipole limit when $M\ne 1$. Keeping $Q{d}^2$ fixed gives a finite pure-quadrupole limit only when $M=1$; otherwise the dipole diverges.

For charges $-Q,2Q,-Q$ at $z=d,0,-d$, the exact force on a charge $-Q$ at $(x,0,0)$ isIt points toward the origin for every nonzero $x$.