Classical Mechanics Level 5

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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