# Relativity

A clever race of Aliens has created an ingenious method of acceleration.

Spaceship 1 travels at $$\frac{1}{2}c$$ relative to their home planet.

Then, spaceship 2 is launched from spaceship 1 and travels at $$\frac{1}{4}c$$ relative to spaceship 1.

This process continues, with spaceship $$n$$ launched from spaceship $$n-1$$ and spaceship $$n$$ travels at $$\left(\frac{1}{2}\right)^nc$$ relative to spaceship $$n-1$$.

Let $$v_n$$ be the velocity of spaceship $$n$$ relative to the home planet. Find $\lim_{n \to \infty}\frac{v_n}{c} .$

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