A visual alternating sum!

This is a short, animated visual proof demonstrating the sum of the infinite geometric series with ratio -1/2.

#mathshorts​ #mathvideo​ #math​ #calculus #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #geometricseries #infiniteseries #fractals #sum #induction

To learn more about animating with manim, check out:

[rushikumar]

The best way is to understand visually. Your channel is so underrated, hope you succeed one day

[JordanMetroidManiac]

Just look at the binary digit expansion of 1/3. Its alternating zeros and ones are of no coincidence.

[anatolykarpov]

It's always great to see your masterpieces, I wish you the best, continue the great work

[edwardwang]

Always admire the combination of algebra and geometry.

[SridharGajendran]

Amazing work.. Can't wait for more..

[rajvlog]

Best channel for learning maths visually so much love from Karnataka, India 🥰

[nathanitet]

this seems similar to the 1/3 infinite alternating sum, in that you add one to the denominator in the fraction, subtract that from 1, at that will be your answer! For 1/2 it's 2/3, and for 1/3 it's 3/4.

[adamdeluca]

Another way, even if it isn't visual: this is a geometric series, so we can calculate the value with the formula for this type of series; so, the result, that I'll call S, will be
S=((-1/2)^∞-1)/((-1/2)-1)=
=-1/(-3/2)=
=2/3.

Anyway, great video!

[VBYTP]

Another way to do it
n = 1 - ½ + ¼ - ⅛ ...
-2n = -2 + 1 - ½ +¼ ...
-2n = -2 + n
-3n = -2
3n = 2
n = ⅔

[asparkdeity]

People can just use the simple geometric formula 1/1-(-0.5) = 1/1.5 = 2/3,

But intuition goes a long way, thanks for the video!

[mr.mrowmusic]

Me just watching a perfect 1 point perspective hallways get made and I can't even see the math anymore

[canadaballplayz]

Another way to understand is A paper (I'm talking A1, A2, A3, A4, A5, A6... yeah it just goes on

[johnpuentes]

Great video, thanks for the knowledge, god bless have a nice day

[JakubS]

1/(1-(-1/2)) = 1/(1+1/2) = 1/(3/2) = 2/3

[gdmathguy]

Here's a way to work this out without visual proofs:
First you can take out the 1. Taking that out makes it possible to use a formula, it'll later be added back
Since it's an alternating sum, we can say that the infinite sum of 1/2²ⁿ minus the sum of 1/2(²ⁿ-¹) would be the same as the infinite sum going from 1 to infinity of ((-1)/2).
Now, 1/2(²ⁿ-¹) is just 2 times 1/2²ⁿ and we can put the 2• outside the infinite sum, so we only have to calculate 1/2²ⁿ and apply it to both infinite sums
1/2²ⁿ is the same as 1/(2²)ⁿ which is 1/4ⁿ. This is where we can use a formula which works with such fractional numbers: infinite sum from 1 to infinity on n of 1/(mⁿ)=1/(m-1). Let's apply that to the 1/4ⁿ and we get 1/3. Now we subtract 2/3 from the 1/3 and we get -1/3 and now we can add back the 1 and it becomes the right solution, 2/3 :)

[QDWhite]

Another easy way to think about these infinite sums is to change base.

E.g., in base 10:

10^0 - 10^-1 + 10^-2 - 10^-3 + 10^-4 - … = 1 - 0.1 + 0.01 - 0.001 + 0.0001 - … = 0.9090… = 10/11

In base 2:

2^0 - 2^-1 + 2^-2 - 2^-3 + … = 1 - 0.1 + 0.01 - 0.001 + … = 0.10101… (base2) = 10/11 (base2) = 2/3 (base10)

i.e.,

x^0 - x^-1 + x^-2 - … = x/(x+1) (baseX)

[johnacetable]

How many more infinite sums can be understood visually?

[eliaparicio]

Can’t you manipulate the alternating harmonic to equal whatever you want?

[dishangdoshi]

you said that we remove and we add and we need to do it infinite times. so does the white one represents the entire part removed leaving 2/3 of the entire shape or the white one is itself 2/3 of the entire. If the white one is 2/3 please explain. Which is the shaded part for you ?

[guleryazkan]

Make a rainbow blocs. be a fractal one🗿