Zeno's spiral into madness

Calculus Level 3

Zeno has just come up with his famous paradox. He begins to panic at the thought of all motion being an illusion, and so starts pacing around, trying desperately to calm his nerves and resolve the paradox.

Starting at \((0,0)\), Zeno goes up \(1\) meter and turns \(90^{\circ}\) clockwise. Each time, the length of the new step decreases to half of its previous step, followed by a turn \(90^{\circ}\) clockwise. So he progresses, \(1\) meter up, \(\frac{1}{2}\) meter right, \(\frac{1}{4}\) meter down...

At the very end, what is the magnitude of the displacement Zeno has travelled from the origin in meters? (Assume that the spiraling takes a finite amount of time.)

Image Credit: Wikimedia Rama
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