Sum of Sequence of Products of Squares of Consecutive Odd Reciprocals

Proof
Note that we have:


 * $\displaystyle \frac 1 {2 j + 1} - \frac 1 {2 j + 3} = \frac 2 {\paren {2 j + 1} \paren {2 j + 3} }$

So: