# 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 $$a+b+c+d$$.

×