Geometry behind the geometric sum of powers of 1/3!

This is a short, animated visual proof demonstrating the sum of the infinite series of the powers of 1/3. #shorts​​ #math​ #calculus #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #geometricseries #series #infiniteseries #fractal

For an alternate series decomposition for the same series, check out

To learn more about animating with manim, check out:

[yusufdenli]

Nice 👍
You can also prove it using circle
1.Draw a circle. Let the area is 1
2.Draw a small circle in this circle with same center. But the area of small circle must be 1/3
3. Divide into two the ring
4.Scan only one
5. Repeat this forever

[wyboo]

could also prove with a power series formula, but this is a lot nicer because it gives intuition

[anjalidwivedi]

Wow literally wow... like a infinite series merged into a simple yet understandable rational solution..
This thing keeps my curiousity about maths growing wild.. 😀😀

[john-icpz]

that's insane 😲
an elegant observation.

[sohamchandane]

My teacher taught me GP and i understood it.
He taught me infinite decreasing GP as well but the derivation was thru the original GP formula and thus there was no actual visualisation of this concept.

Ur videos really help us visualize maths in a completely unique way❤

[joshyoung]

This is my third of these videos I've seen, and I haven't played with visual proofs or series in a while but I'm starting to understand why you can come up with so many of them. First off, you can do basically any of them using the technique you used in your sum of powers of 1/8 video with the heptagon, but for a lot of examples, you can do other fun tricks like this that make really cool-looking and easily-understoof proofs.

[SoundsOfTheWildYT]

As a corollary of this, the representation of 1/2 in base 3 is 0.111…, which also gives that 0.222… = 1 (equivalent of 0.999… in base 10)

[sharpnova]

for those that didn't get it, it's one half because you create two spirals, the shaded one and the complement of that, they are both the same

[vladimirrodriguez]

Great animation that shows the convergence experienced by this geometric serie (its ratio is 1/3 <1) with its final value being 1/2 (according to the formula it is: first series term/(1-reason)).

[thebarch]

express as the sum from 1 to infinity of 1/3*(1/3)^(n-1) and since it converges it equals (1/3)/(1-(1/3)) which equals 1/2

[azvil]

Can be done by GP
a=1/3
r=1/3
n=infinity
S=a(1-r^n)/1-r
=1/3(1-1/3^n)/1-1/3
= 1/3(1-0)/2/3 as, (.33)^ infinity=0
=(1/3)/(2/3)
=1/2

[znhait]

With something like this, it’s a lot easier to explain and understand it AFTER learning about power series and geometric series. Once you know the answer, you just work backward and find any interpretation that is consistent.

[inskeeprulerable]

I'M SO EXCITED! I watched the infinite sum videos for the powers of 1/4 and then the powers of 1/2. I noticed that there seemed to be a pattern (infinite sum of powers of 1/4 = 1/3, and infinite sum of powers of 1/2 = 1) so I figured the infinite sum of the powers of 1/3 must be 1/2!

So cool.

So does that mean the invite sum of the powers of 1/5 = 1/4???

[rooblixkewb]

I'm seeing a pattern.
(1/3) + (1/3)² + (1/3)³... = 1/2
(1/4) + (1/4)² + (1/4)³... = 1/3
(1/5) + (1/5)² + (1/5)³... = 1/4
Can someone give me some answers? Does this sort of pattern work with all fractions, or are there specific fractions it doesn't work on?

[Dissimulate]

From the formula for a geometric series with the ratio on the interval (0, 1), 1/(1-(1/3)) - 1 = 3/2 - 1 = 1/2

[Fishinator]

the visualization is great! as for the actual maths, it got me thinking (this and other videos)
is the 1/n geometric series always 1/n-1?? as in 1/3 geometric series is 1/2 and so on. you can use geometric infinite series formula to prove it pretty easily

[Kornithe]

Golden Spin ?




Is that a jojo reference?
(Yes i know it scale 16:9, not 2:3 )

[Xfady]

Man, You are my mathematical hero 🎉🎉🎉

[shriomomar]

love ur videos broo u should be appreciated

[robusk]

This was well thought and well done.
Aside from daily jobs, none would of ever thought about this.
This is pure beauty