Arithmetico-Geometric Series Visualized (visual proof using Gabriel's Staircase)

This is a short, animated visual proof computing the sum of the differentiated geometric series with terms of the form k times r^k where r is between 0 and 1. The proof uses a fascinating object called Gabriel's staircase, which has infinite perimeter but encloses a finite area. #math​ #infiniteseries #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #calculus #geometricseries #series #mathshorts​ #mathvideo​

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[BrianAmedee]

This is an amazing abstraction. Awesome job!

[DitDede]

Nice. Alternatively (not a visual proof), the left sum is the derivative of r/(1-r)