Multipole Expansion Intuition So my intuition of the multipole expansion is that we have a certain charge distribution, and the ME tells us that we can approximate that distribution as a superposition of monopoles, dipoles, etc. Physically we could have a certain dipole for instance, then adding a quadropole to this takes us a step closer to the original charge distribution. Repeat this an infinite number of times, adding higher order terms like octopoles, 32-poles etc and we would end up with the exact charge distribution. Correct me if I am wrong, but would this be the intuition behind ME?
 A: Yes, that's correct. It is similar to the Fourier series where a periodic function is rebuilt using a basis of oscillating functions. The Fourier decomposition translates a periodic function of one continuous variable into a function of a discrete variable (an index).
If you consider a function of the two angles $\vartheta$ and $\varphi$, that is, a function whose argument is a direction pointing somewhere on a unit sphere, you can use the spherical harmonics as a basis. Here we have a function which is again periodic in its two variables and translate it into a function of two indices. This is used, for example, when analysing the cosmic microwave background which is described by a function $T(\vartheta, \varphi)$ giving the temperature of the background radiation depending on the direction in the night sky, you are pointing your detector at. You can get a set of coefficients $a_{lm}$ from a decomposition in spherical harmonics $Y^m_l(\vartheta,\varphi)$.
If you are interested in a function which is defined on whole $\mathbb{R}^3$, the different $l$-modes scale with some particular power of $r$. But this is specific to the particular function we are expanding, typically the function $\frac{1}{\vec r-\vec r'}$ which is the Green's function of the Laplace operator.
