Infinite tangents

Calculus Level 5

Consider a point \(P_1\) on the curve \(y=x^3\) such that the tangent on \(P_1 = (1,1)\) meets the curve again at \(P_2\). And the tangent at \(P_2\) meets the curve at \(P_3\) and so on.

Let \((x_n,y_n)\) be the coordinates of \(P_n\) then evaluate:

\[ \displaystyle \dfrac{\displaystyle \lim_{n \to \infty} \sum_{r=1}^{n} \dfrac{1}{x_r}}{\displaystyle \lim_{n \to \infty} \sum_{r=1}^{n}\dfrac{1}{y_r}}\]

If the answer is of the form \(\dfrac AB\), where \(A\) and \(B\) are coprime positive integers, find \(A+B\)..

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