For arbitrary $r_0$ and $P_l(\text{cos}(\theta))$ be the Legendre polynomials,
$$ E_n=\int_{r0}^\infty \int_0^\pi -\text{sin}^3(\theta)(\left( \left(\text{cot}(\theta) \sum_{l=0}^n R_l(r) \partial_\theta P_l(\text{cos}(\theta))\right) + r \left(\sum_{l=0}^n R'_l(r) P_l(\text{cos}(\theta))\right)\right)^2\text{d}r\text{d}\theta $$
(monotonically) decreases as $n$ increases. i.e. I want to prove that the sequence $|E_n|$ must be decreasing.