Step 1: A point charge q sits at distance a on the z-axis. Find the leading terms of the potential far away (r >> a) using Legendre polynomials. Step 2: The exact potential is V = (1/4πε₀)·q/√(r² + a² − 2ar·cos θ), where θ is the angle from the z-axis. Step 3: For r >> a, factor out r: V = (q/4πε₀r)·1/√(1 − 2(a/r)cos θ + (a/r)²). Step 4: The factor 1/√(1 − 2xt + t²) is the generating function of Legendre polynomials: it equals Σ_l P_l(x)t^l with x = cos θ and t = a/r. Step 5: Keeping l = 0 and l = 1: V ≈ (q/4πε₀r)[P_0(cos θ) + (a/r)P_1(cos θ)] = (q/4πε₀r)[1 + (a·cos θ)/r]. Result: The l = 0 term is the monopole potential q/4πε₀r; the l = 1 term is the dipole correction proportional to cos θ.