Infinite sum of powers of sixths!

This is a short, animated visual proof showing the sum of the infinite geometric series of powers of 1/6 (starting with 1/6) is 1/5 using a general method of dissecting a circle into a small circle and annular sectors.

To learn more about animating with manim, check out:

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

[alekhon]

This method with circles works for any series with powers of fractions like 1/n

[robusk]

My brain has gotten a new dimension of visualising APs & GPs. Thanks to your amazing content.

[quantumgaming]

You sure do love the geometric series, sir xD

[chilldo]

I feel like it's proven that the sum of powers of any fraction, equals the fraction with the denomitator one smaller. Or at least when the top part is 1.

So the sum of powers of 1/n is 1/(n-1)

[thekrackedbush]

Thankyou so much this knowledge would help me a lot in studying AP and HP ❤

[InfestedSlab]

brain.exe stopped working
reboot initializing.

[bettercalldelta]

If I remember correctly, the sum of powers 1/n always equals 1/n-1

[kokilabhuptani]

This has a really simple formula
A/(1-r)
A is the first term of the series
R is the ratio you divide the terms
Here A=1/6
r=1/6
So the ans would be
(1/6) /(1-1/6)
=(1/6) /(5/6)
=1/5

[MrDeyzel]

I like the new narration approach. Great channel.

[AngryEgg]

The sum of any geometric sequence where a1 is 1/n and also equal to q is 1/(n-1)

Proof:

Sum of infinite geometric sequence:
a1/(1-q) => (1/n)/(1-(1/n)) = (1/n)/((n-1)/n) = 1/(n-1)

[the_m_original]

let x be any number you'd like, and construct a sigma notation:

∞
E (1/x)^n
n=1


the result will always be 1/(x-1)

[johnsterling]

this video and the other one on 1/3 really helps me understand these summation type problems. could you throw in a variable and show us how it affects it???

[malicacidissour]

If you were my math teacher in Highschool, then Maths would’ve been a much cooler subject.

[KaliFissure]

Powers of 6 gives a fifth
Powers of fifths give a fourth
Powers of four give a third...

Children need to learn this fairly young because it is so simple and elegant yet somehow mysterious

[limenlemon]

Try this method with (n-1)/n. It will technically be one, but the circle will never be completely filled.

Also, for the powers, use (n-1)/(n^m) instead of ((n-1)/n)^m.

[williamhenby]

This same process also works if you use a regular pentagon

[capoa]

We do love the infinite sum of a geometric series though lol :)

[richardchristie]

Love these. Makes something click in my brain!

[anjalidwivedi]

it's always the same formula
sum of powers of 1/n = (1/(n-1))

[raymondfrye]

So the área of the circle will be one less the denominator of the fraction.