Proof by intuition done by Leonhard Euler, sum of 1/n^2, (feat. Max)
ฝัง
- เผยแพร่เมื่อ 18 ก.ย. 2017
- Check out Max's channel: • Introduction to Totien...
Check out Fouier's way, by Dr. Peyam: • Video
Sum of 1/n^2,
pi^2/6,
blackpenredpen,
math for fun,
![ป่ะป๊าเลอา ชื่อ ??? [ดูคลิปเต็มหน้าช่อง] #Short | PANG ORNHIRA](http://i.ytimg.com/vi/ssxX0VkJCTo/mqdefault.jpg)
To show b=1, just compare the constant terms in the two infinite-nomials
Yeah, simple! I love algebra:)
It should be β=1
when you fiddle with the coefficient by x^4, you'll get sum of 1/n^4 = pi^4/90
Yeah it’s amazing, you can do this with every even nth power!
5:38 to skip the intro to polynomials and roots
Take limit as x->0 of both sides. Sinx/x approaches 1 (well known limit) and right side approaches beta*(1*1*1*1*...)=beta. So beta=1.
could someone maybe explain to me why multiplying an infinite number of 1's here approaches 1, instead of being an indeterminate form?
Ronald K. This, however, is not the same with the limit we got here. In the limit, the infinite factors are already there, they are not infinite because of the limit. I hope you understand what I am trying to say :)
The actual problem is multiplying infinitely many things. You could in Theory take the limit of a finite product but then you'd have to prove that these finite products actually approach sin x which is not that obvious i think. Basically this whole proof is not actually rigorous but it makes sense intuitively and gets the correct result so you jost gotta act as if multiplying these infinitely many things is fine.
Ronald K. yes, that's basically it
Sharon Y. I was about to say the same thing. But not with the limit, just to set x=0.
sin(x) isnt a polynomial but the Taylor series of it is a polynomial
Oon Han But if the Taylor series is an exact representation of the function then isn't it the same thing?
The Taylor series is only an *infinite* "polynomial". Like a polynomial is only a *finite* "series".
Sin(x) is only an *infinite* "polynomial", that is, a series.
I would probably call it power series, rather than a polynomial. Polynomials are defined as finite for Modern Algebra reasons.
A polynomial has a finite degree. That’s why pi is trancendental even though it is the minimum positive real root of the series expansion for the sine function
@@janderson2709 an infinite polynomial is a series not a polynomial
Recommendation: dude, if you are making a video about complex numbers, nth term sums and taylor series. obviously the audience knows about polynomials. explain everything or not explain.
nice video!
No not neccesarily. I'd you've taken calc II then you know about complex numbers, infinite sums, and taylor series. The polynomial stuff is Modern Algebra. And besides all of that is over the lay person's head. Explaining is good.
Poor me, I've had calculus 20 years ago. It did good to me to brush up these in a second.
I just love your channel, thx so much for sharing all these cool videos. Cheers from Brazil
That was very elegant!!! It bridges algebra with analysis ... I like that ... He was all over the place ... but he will get better for sure ...
It it does not terminate,, it is not a polynomial - it is a power series!
I like this version more, as I don't have deal with some fancy theorem that I don't understand
David Franco it's a theorem: Basel problem
I had
The thoerem here is that sin(x) actually equals that product, which is harder than the other theorem
what fancy theorem?
I watched this video so much that I just love the 'nice' at the end of the video.
“Proof by intuition” gives the same vibe as using “this was once revealed to me in a dream” as a citation
Very very very great and nice solutions that I have ever seen. thank you very much for your presentation.
Some years ago, I found a proof that Pi²/6 is the same as:
e^(Σ_n Σ_p 1/(n*p^2n)) where 'n' are integers 1 to ∞, and 'p' are primes.
I think Euler did the same, but he had a lot more time ^^
I'd like to have friends like that some day, nice video
I like this démonstration
I like your team
Multiply everything by pi, and then the sum is a bunch of circles with radius 1/n, and for n is all integers you get a cone like shape, whose area is proportional to pi*h/x, for h is the largest value of n, and x is something dependant on n... It ends up as h/x = pi^2/6
Wonderful!
Beautiful proof! The three of you are grea
Very impressiv!
totally amazing......something that I can understand
great proof sir wonderful thing beatiful!!!
@blackpenredpen There's a typo in the title. His name was "Leonhard", not "Leonard". :)
Thanks!!! I just fixed it!
Thanks for your video!
Nicely presented but I share the same concerns with some of the viewers. Is it allowed to extend a theorem which applies to a finite number of elements to an infinite series?
Two cases:
1. Factorization of an infinite polynomial (Taylor expansion of sinx)
2. The assumption that the limit of a sum of infinite functions equals to the sum of their limits (provided these each of these limits is finite of course).
Awesome video. Taking limits on both sides makes B=1. Cool!
He is a beast
Great sir thanks a lot
These are genius of maths.
Wow man what a approach 😱😱😱😱
great..thanks!!!
Молодец. Отличное доказательство. Теперь осталось это сделать через ряды Фурье .
A huge clap for this guy.
I understand the idea of taking the limit of both sides, but I'm not sure about the result (maybe on the right side we have a (infinity)*0 form? Where (infinity) is given by Beta, and 0 by the product of infinite "0.9999..." terms)... let's consider what you wrote:
sin(x)=x(x^2-pi^2)(x^2-4pi^2)(x^2-9pi^2)...
So:
sin(x)/x=(x^2-pi^2)(x^2-4pi^2)(x^2-9pi^2)..., then:
sin(x)/x=(pi^2)*(x^2/pi^2-1)(x^2-4pi^2)(x^2-9pi^2)...
In other words, I wrote (x^2-pi^2) as (pi^2)(x^2/pi^2-1). I can repeat the same operation with the other factors, so we have:
sin(x)/x=(pi^2)(x^2/pi^2-1)*(4pi^2)(x^2/4(pi^2)-1)*(9pi^2)(x^2/(9pi^2)-1)...
Rearranging the factors on the right side, we get:
sin(x)/x=(pi^2)(4pi^2)(9pi^2)...*(x^2/pi^2-1)(x^2/4(pi^2)-1)(x^2/(9pi^2)-1)...
So, shouldn't your Beta be equal to (pi^2)(4*pi^2)(9*pi^2)(16*pi^2)...?
Have you heard about conditional convergent series? On the right hand side, you deal with infinite number of terms, so we need to be careful. I think, I did not explain well the transition that I made in decomposition of finite polynomial. I want to make a short talk when I will show this step. Also, you discovered one of the paradoxes:) so that's the reason why this is not a rigorous proof.
Thanks
For the issue looking at the constant B does this make sense to anyone?
Note I’m using xn in place of n^2Pi^2
Keeping in mind that a polynomial can be factored as P(x)=B(x - x1)(x - x2)…(x - xn)
But in this form THE INFINITE PRODUCT does not converge. In the video the constant is left out for the infinite product for sin(x) when the instructor forms the polynomial after the Taylor series.
Bx(x − x1)(x − x2)(x − x3) . . . (x - xn) n → ∞.
The answer is no, because (x − xn) → ∞.
However we can modify it in the following way: factoring each xn and modifying B to B′
That is just dividing each factor on xn, and adjusting the constant B in front accordingly.
The first infinite product does not converge but the second does:
B’x(1 - x/x1)(1 - x/x2)…(1 - x/xn) n → ∞ converges
So the first B is undefined but B’ can be found by dividing both sides by x and looking at the limit of sin(x)/x where x -> 0.
Also, given that you can determine beta=1, this allows you to prove lim as x approaches 0 of sinx/x=1 if you didn't know it already.
Awesome
Nothing against the guest speakers, but I'd prefer less of them and more of blackpenredpen himself. Interesting proof, though.
Camera in front the board !
So we can see and understand something !
Can Beta not be represented as the product of (-(n^2 * pi^2)) from 1 to infinity? I am not sure how it is resulting in 1.
Me neither!
β is a constant term, but on the RHS, the constant term is 1, so β = 1
forget the limit....u can compare the coefficients of x^0 from the formed eqs.
actually we can compare the coefficient of constant term so beta(1)(1)(1)...=1 ,so beta=1
I guess you know the problem that Euler's proof faces. Making a conjecture that is true generally for finite for the case of infinite. I was wondering if you could do a video on Cauchy's proof which uses the sandwich criterion for limits... @Blackpenredpen
The camera placed is in the wrong way that could harm the vision of the onlookers
6:20 subtitles: "they have fine fat goose".
Just take the limit as x approaches zero of both sides. sinx/x simplifies to 1, so β = 1
Great
I don't understand this passage: 10:40. can you help me, please????
Do int csc x dx the standard way and this way: int csc x dx= int (sin^2+cos^2)/sin x dx= int sin x dx + int cos^2/sin x dx. Use u= cos x for second part, partial fraction
At 8:10, sinx = infinite factors it is correct, but a constant must be mutliplied like insteas of x(x-pi)(x+pi) ... it should be beta.x.(x-pi)(x+pi)... etc
6:33 You don’t even need to consider the Complex numbers realm; even on the Cartesian (real) »x« axis, sin(x) »crosses« that axis infinitely often periodically, all the way from -∞ through 0 through +∞. When I was introduced to trigomometric functions in German »Telekolleg« television, I first thought, all these repetitions, what a waste (ha!), but soon after recognized their beauty and utility. Cheers!
I like it
very cool indeed.
You can show that betta is equal to 1 taking the limit when x goes to 0.
Superb bro🙂🙂
That is quite a handy method for calculating all even Zeta values.
Well done.I felt this guy explained better than the guy wearing specs who is there in most videos.
11:17 why in the coef of x^2 there is no a betha
whats funny is we literally just talked about this in class, but we did the fourier way
Dr. Peyam did that too: th-cam.com/video/erfJnEsr89w/w-d-xo.html
so sinx is well approximated by an infinite polynomial via Taylor .. and the same polynomial most be represented by an infinite product of linear terms that specify the roots of the function (just need to find the citation, Weierstrass factorization theorem ?) If I can accept that then the rest is just busy work.
Really nice proof. Did you discover it yourself? The whole representing the Taylor series as an infinite product reminds me of the Euler formula for the Riemann zeta function. Are you by any change a researcher in analytic number theory? The only other proof I have seen of this result apart from the Fourier series type approach uses contour integration and the residue theorem.
Shaun De Roza This proof is actually due to Euler. However it wasn't acknowledged as one for the non-rigorous methods he used (classic Euler) until Weierstrass completed it. Apparently the numerical confirmation convinced him to publish his results.
I am guessing he couldn't be bothered formalising the notion of infinite products or proving that expanding sinx as an infinite product converges to sinx pointwise.
Thanks! Actually, it's due to Euler. I read about I a couple of years ago. No, I actually do research in algebra and algebraic topology, but I am really interested in this kind of things.
For a cal 2 student, it's hilarious to see "is sinx a polynomial?" on the board XD
I dindt understand what did max do in theend
How to prove the maclaurin series for sinx
Take a bunch of derivatives and then plug into the formula :)...since it's Maclaurin c=0
No one bothered by the multiplication of an Infinite amount of factors and taking the separate limit of each one?
This would prove that lim (1+1/n)^n = 1 ...
I understand that you are vulgarizing, but a disclaimer about the non rigorous aspect would be nice
Yeah, I like you question. I think in that case n is in the base and in the power, so you cannot split it into the product. Take a look at the proof that the limit of product is the product of limit. In my case, I have my x only in the base.
Well, I agree that the growing list of factors is constant (only one new factor for each new product), so we need a proof on a finite product and then make it tend to the inf. But I still find this a little too quick. ;p
Skycoca No, it is not valid to split it into a product if the exponent is also a variable of the limit. This is not unrigorous at all, you’re simply making a mistake. In reality, the limit distributed here onto the exponent and base, not onto factors.
sin(x)/x can be made continuous at 0 by defining the quotient to be 1 at 0, and the polynomial on the right (in the video) is perfectly defined, it is just an infinite product of 1, and not an infinite product of things that converge to 1.
The limits in multiplication can be seperated if they exist individually and finitely. And they do in this case.
3:27 Shouldn't it be "every non-constant polynomial" there? (degree > 0). p(x) = a0 has no roots unless a0 = 0.
Waaaoooooooo!!!!!!
If you just expand the polynomial on the LHS the first term is clearly 1 multiplied by itself infinitely many times and multiplied by beta. The RHS indicates that the first term should equal 1, therefore beta*1=1 and beta = 1
That Taylor series is valid only up to one radian, much less than pi. Interesting that it works out anyway.
Family of the Euler perhaps? 😂
Around 4:56, he forgot to mention that alpha 1, alpha 2 etc couldn't be zero as he is dividing by them. So it is true for non-zero roots.
"The proof of beta=1 is so trivial, its proof is left to the viewer as an exercise"
Peyman is so clever
Is that guy russian? His pronunciation sounds like russian
sin(x) not eq x*(x^2-pi^2)*... check x=pi/2 you get infinity!
Fourier Series😀
Mathologer did a nice video on this same proof. :D
Am I on wrong channel?
a little tip, next time move the camera to show his notes on the board rather than getting a angle view.
I have many questions though
Can anyone answer
First
In the first expression
sinx/x=(x-π^2)(x-4π^2)......
Here if we compare the coefficients
we get something weird
Secondly
What happened to β while comparing the coefficients
Why is (x² - pi²) divided by x² equal (1 - x²/pi²) rather than (1 - pi²/x²)?
Factor constant term
he divided by negative pi squared not by x squared
The alpha coefficient was left out from sinx = x(x^2-Pi^2)(x^2-4*Pi^2).... and
how explain tg x = x+x^3/3 + ... and tg x = 0 eq sin x = 0. I can build similar product but sin x not eq tg x.
I understood this... the parseval thm one was hard...
What is Max's last name? What is his bio?
beta is obviously 1, if you take the limit as x goes to 0 on both side
Is Max a PhD student?
Someone please explain why the chain rule for d/dx of xx/x and 1/(1/x) don't work (no simplifying allowed)
It totally works.
Who said it does not?
It is not the chain rule that fails, but the function itself, and hence all of its derivatives. By making these rewrites you have 'punctured a hole' in the function at x=0, where it is no longer defined. It is trivial to 'backfill the hole', but you disallowed simplification, so you are stuck with your hole at x=0 where nothing is defined.
if you set x=pi/2 you get 2/pi = b* product n=1 to infinity (1-1/4n^2)
La pizarra de atras esta en español :O
Quick limit limits comments!
I will call the value of finding beta as a paradox because mathematically it can be proven but logically it seems something weird please can any one help me
No 1^2+2^2+3^2+...+n^2+..=0 a well known result in fact 1^(2m)+2^(2m)+3^(2m)+....+n^(2m)+...=0 for all positive integers m.
He looks like Jesse of the Braking Bad
Hello math lover a have a proof and also a theorem in finding the higher order derivatives of any nth root polynomial that can evaluate in just 3 to 5 seconds. Please recognize.
It's ok
Man you forgot to show B= 1 in the expansion of sinx/x
Mathematics work!!!!!
Please use the microphone next time! :)
Sure. Thanks)
so.... (-pi^2)*(-4pi^2)*(-9pi^2).....=1?
raziel Keren where did you get that from?
to get betta in the front you have to factor out thos terms above.
since by limits we have to say beeta=1...
You have to account for the infinite terms in the product of the roots. That's where the magic happens.
raziel Keren yeah i don't understand either
its the same magic like 1-2+3-4.....=1/4
it's going frpm -inf to +inf so in some connection we can say there is a value to the sum.
and this is true in the same way 1+2+3+4... = -1/12
but the identity in the video is correct in any connection so the conclusion is that I'm missing somthing.