Sum of Sequence of Products of Squares of 3 Consecutive Reciprocals

Proof
We use partial fraction expansion to expand $\dfrac 1 {j^2 \paren {j + 1}^2 \paren {j + 2}^2}$.

Let $\dfrac 1 {j^2 \paren {j + 1}^2 \paren {j + 2}^2} = \dfrac A j + \dfrac B {j^2} + \dfrac C {j + 1} + \dfrac D {\paren {j + 1}^2} + \dfrac E {j + 2} + \dfrac F {\paren {j + 2}^2}$.

This has solutions $A = - \dfrac 3 4, B = \dfrac 1 4, C = 0, D = 1, E = \dfrac 3 4, F = \dfrac 1 4$.

Thus: