Working with Cubic Polynomial

Algebra Level 5

For a positive integer \(k\geq 2\), let \(a_k,b_k,c_k\) be the complex roots of the equation \[\left(x-\dfrac{1}{k-1}\right)\left(x-\dfrac{1}{k}\right)\left(x-\dfrac{1}{k+1}\right)=\dfrac{1}{k}\] and let \[p_k=a_k(b_k+1), q_k = b_k(c_k+1)\text{ and } r_k = c_k(a_k+1).\] Given that \[\sum_{k=2}^\infty \dfrac{p_kq_kr_k}{k+1} = \dfrac{m}{n}\] for some coprime positive integers \(m\) and \(n\), find \(m+n\).

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