# Parabola, tangents and superficies

Geometry Level 5

Consider a parabola $$P$$ given by $$(x, x^2)$$, with $$x \in \mathbb{R}$$, and $$(a_n)_{n \ge 0}$$ a sequence given by $$a_n = n^\beta$$, with $$\beta > 0 \in \mathbb{R}$$. Consider the superficie $$S$$ given by the region at the right side of the vertical axis ($$x=0$$), below the parabola $$P$$ and above the lines tangents to the parabola $$P$$ on the points $$(a_n, a_n^2)$$.

There is a interval $$0 < \beta < \dfrac{M}{N}$$ which makes the area of $$S$$ be finite. What is the value of $$M + N$$?

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