# 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 $$A+B$$..

×