Given $\sum\limits_{r=1}^{N}{\frac{1}{4{{r}^{2}}-1}}=\frac{N}{2N+1}$, Deduce an expression for $\sum

Question

Given $\sum\limits_{r=1}^{N}{\frac{1}{4{{r}^{2}}-1}}=\frac{N}{2N+1}$, Deduce an expression for $\sum\limits_{r=1}^{N}{\frac{1}{(2r+1)(2r+3)}}$.

Accepted Answer

N/(3(2N+3))

Worked Example 6