# An infinite product inside a series

**Calculus**Level 5

For each positive integer \(m\) we define the function \[P(m):=\prod_{k=1}^\infty\left(1-\dfrac{(2m)^2}{(2k-1)^2}\right).\] If \[\sum_{m=1}^\infty P(m)\dfrac{m^2+2m-1}{m^2-2m^3}=\dfrac ab \pi+\dfrac cd \pi^2\] for some positive integers \(a,b,c\) and \(d\) such that \(\gcd(a,b)=\gcd(c,d)=1\), find \(a+b+c+d\).