Visual proof of infinite geometric series sum II

This is a short, animated visual proof demonstrating the infinite geometric series formula for any positive ratio r with r less than 1 and with positive first term a. This series is important for many results in calculus, discrete mathematics, and combinatorics. #mathshorts​ #mathvideo​ #math​ #calculus #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries

If you like this content, please consider subscribing.

To learn more about animating with manim, check out:

______________________________
Music in this video:

[DFSkingA]

Wish I had math knowledge like you my videos could go to the next level

[randmsfake]

(I feel like I'm becoming regular nitpicker... but) It did feet weird to me that Triangle Similarity was used as proof after purple line was drawn on the video. I think it would have been better to compose purple line itself with Triangle Similarity as that would prove that it is a straight line.
Also I'm tad puzzled why height and and widht are scaled differently (unless a=1). At least I myself got rather confused by it, as since those blue rectangles look like squares, I kept (incorrectly) thinking that "a" was supposed to be 1 and that would somehow be important.
Otherwise a nice video.

[Adaptibility]

I am confused why is the bottom a and ar when those are the areas of the rectangles themselves.

[dre]

... and dividing both sides of the result by "a" removes "a".