Alternating Geometric Series Sum

This is a short, animated visual proof demonstrating the infinite alternating 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.
Thanks!

#mathshorts​ #mathvideo​ #math​ #calculus #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries

To learn more about animating with manim, check out:

[ZeMeatMuncher]

When people say "math is beautiful, " things like this are what they mean.

[raymitchell]

The isosceles triangle is so underrated LOL

[HermesAngel]

I'll be really honest. I have no idea what you're talking about, but I like listening to you talk about it

[DanielPoupko]

That was your best one yet. Brilliant.

[andrewkarsten]

My analysis book had a picture of this for the cover

[jedsonguedes]

Probably the best visual proof of this channel so far!

[virushk]

You sir are a genius, this explanation was a masterclass

[randomproductionscj]

My first thought was - who tf uses y=-rx+a rther than y=-mx+c

[jamium]

Does this correspond to r being a negative number? This is a beautiful animation, is it manim

[vennstudios]

Basically

Summation to infinity n=0 of
A×(-r)^n
§ is stand in for summation because no symbol in keypad :(
Or just Aק-r^n

[.DiepAnhTuan-bxcg]

Good good thanks for the information brother

[RenneVangr]

I understand it, but it still messes with my brain

[Gabriel_Bento]

Are those animations made in Python (Manim)?

[mosescheung]

i can't see why the distance between the purple line with length ar and the line y=-rx+a is ar^2, can anyone help

[pizza]

Substituit u=-r and you will get a+au+au²+....=a÷(1-u)

[rahulsai]

How bro how you imagine all this stuff 😮😂😂

[reedmiller]

The piece I’m unclear about is why we can be sure it’s an isosceles triangle if the absolute value of the slopes of the lines ( +1 and <-1 ) aren’t equal ( 1 and <1)

In this image r looks like it’s approximately 1, what if r were, say, ?

[jasonc]

All i know is a and r are lines at an x coordinate so yeah im smart

[milesfreilich]

No, it would be (a/1+r)/2 since the final result you showed was half