# 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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