Visual Harmonic Series Divergence from Bernoulli!

This is a short, animated visual proof showing Bernoulli's method of determining that the harmonic series diverges. This technique is similar to, but different from, the technique used by Oresme. #manim #math #mathvideo #series #infiniteseries #mathshorts #geometry #animation #theorem #pww #proofwithoutwords #visualproof #proof #area #harmonicseries #pww​​ ​ #proof​ #areas #mathematics​​ #mtbos #harmonic #divergence #bernoulli #oresme

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[elidoz]

the best proof I've ever seen of this

[adrianv.v.]

Had never seen this proof, beautiful. 👌

[AlisterCountel]

I usually think of the powers of 2 trick, but this is also a neat proof!

[mohammedal-haddad]

This is so clever. Thank you very much.

[Ninja]

Another way would be to say that the harmonic series is bigger than the area under 1/x from 1 to inf by drawing rectangles, similar to what you did. But since the area, which is really the integral of 1/x from 1 to inf diverges, the harmonic series must also diverge.

[Shaeffen_]

Proven to be an infinate sum of 1s even though it seems to be less than 2 😵‍💫

[YOUTUBE_AMERICA]

Love from India you are better than my teacher

[eliasboudjella]

The last inequalities seems to be wrong because ln(n)<n

[kevinbihari]

Sorry man, you lost me
Why the graph? What is c?
Why 1 over c squared?
Hiw do you know what the y coordinate is at x=c squared if c is just a fantasy number?
How does that get applied more than 1 times?

I really love your style. Would you make a longer video please?

[maxbow-arrow]

Great proof! Here's another one:
We can replace every element of the harmonic series with a smaller power of two (for example, 1/3 -> 1/4, 1/5 -> 1/8 etc.). Then our sum becomes which is just an sum of infinite halves. This sum is smaller than the harmonic series, so it also diverges.

[ThePeterDislikeShow]

I've always learned it as powers of 2.

[MochiClips]

I like the 1/2 explanation

1>=1
1/2>=1/2
1/3+1/4 >= 1/4+1/4=1/2

1/5+1/6+1/7+1/8>= 4*1/8 etc...
Doubling the number of points and halving the denominator getting an unbounded sum and hence divergent 🗿

I like your argument though it's nice 😎

[JHnat]

You are going to eventually get to 2. 1+ 1/2… so on and so forth. It will approach if not reach 2

[Mike-nvwz]

you used the associative property to rearrange the terms, partial sums. it is valid if the series is regular ( divergent or convergent) . how do we know the armonic series is regular? i miss this point

[Hassan_MM.]

How does c & c^2 fall on x axis & plz do explain Height of Bottom Rectangle❤

[victormagaud]

can't we just use the fact that lim(sum(1/k))=integral of ln(x) between 1 and +infinity which diverge ?

[單邊襪]

As a person who never touched calc Im kind of confused, why cant we just pair up each component of the two series and say that all 1s are bigger than the other half / third / quarter, so the infinite sum of 1 should be bigger?

[TryJesusNotMy]

Wow answer is infinity how didn't I guessed?😮

[beath-yy]

But you cant devide infinite series by parts of sums if it is diverges

[feyindecay]

Sadly I don't understand... does this have to do with music theory?