Two Infinite Series Sums from Regular Polygons (visual proof)

This is a short, animated visual proof computing the sums of two series - one of reciprocals of triangular numbers (i.e., certain binomial coefficients) and the other a classic series that is used to demonstrate telescoping series.

For some related videos see these videos:

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[ND]

I mean, if you wanna get really technical, the limiting shape when you increase sides of a regular polygon isn’t a circle, but an Apeirogon. Though if you could see an Apeirogon in its entirety it would be indistinguishable from a circle, so in the end it doesn’t change the final result

[PRIYANSH_SUTHAR]

I have no words to describe my wonder at this.

[cosmicvoidtree]

Well, in a sense for the visual proof, we recognized the telescoping series as the one we originally wanted to find

[fedorshestirko]

What is the name of the program you use to paint?

[wieneryron]

Is the limiting circle really that much larger? I would not have expected...

[Dyslexic-Artist-Theory-on-Time]

Could an emergence process of spherical symmetry, represented by 4πr², forming and breaking, shape the characteristics of three-dimensional space by creating statistical entropy with the potential for ever-greater symmetry, similar to cellular life? Photons with energy ∆E=hf continuously transform potential energy into the kinetic energy of matter, Eₖ=½mv², in the form of electrons. Could this process establish a design pattern or template based on spherical geometry that allows for self-organization and the emergence of complexity?

[LooWoo-pmuk]

Hello, can I buy the source code in one of your videos, because I want to learn it, can you, for example, the source code of the intersecting string theorem, I like your videos very much

[dyneek]

I don’t understand why you said the sum is equal to pi. Wouldn’t it just approach pi since the shape isn’t really a line, and does have curvature

[daniel-fvrh]

I'm learning algebra 1 myself by watching videos i do problems
I understand many some not I'd have any maths book shelf guide anything like that i just downloaded a pdf of algebra self study but i can't relate to that I have studied in videos (why is that ) I'm 19

[SteveThePster]

Not 100% if this is rigorous as a proof - seeing as there is no limiting shape (as you cannot have an infinitely large circle).
To turn it into a proof, I imagine all you have to do is prove that the limit (of the numerical series) exists

[IainDavies-zl]

They go on about this TOSH, and can't even tell what 1/76 scale weight of 126t is.