rr = Sqrt[r^2 + rp^2 -   2 r rp Cos[θ]]

series = 
      Series[1/rr, {r, ∞, 7}]   // FullSimplify

Table[LegendreP[n, Cos[θ]], {n, 0, 6}]  // Together // 
      TableForm

Table[
      SeriesCoefficient[series, n + 1]/LegendreP[n, Cos[θ]]  // 
        Simplify, {n, 0, 7}]