Cauchy's Proof of the Basel Problem | Pi Squared Over Six (3blue1brown SoME1 Entry)
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- เผยแพร่เมื่อ 22 มิ.ย. 2024
- Cauchy's Proof of the Basel Problem | Pi Squared Over Six (3blue1brown SoME1 Entry) // If you're looking for challenging or tricky math problems, you found the right video. The Basel Problem was one of the most challenging math questions that stumped many great mathematicians, including the famed Bernoulli brothers.
The first proof was provided by the great Leonhard Euler in 1734. The solution is pi squared over six - a surprising "pi formula" which gets the exact sum of the reciprocals of the squares of all the natural numbers from one to infinity. Whereas Euler's proof requires elements of calculus, the proof by Cauchy, which we explore in this video, only requires advanced precalculus. Some necessary ingredients to complete the proof include the binomial theorem (expansion), De Moivre's formula, the squeeze theorem, and Vieta's formula. Plus a lot of trigonometry (grade 12), trigonometric identities, and algebra.
Other mathematical proofs also exist and we need look no further than our fellow math channel friends on TH-cam for their awesome explanations. A geometric proof of the Basel Problem is given by 3b1b and an awesome calculus based proof in the complex domain is given by Black Pen Red Pen. Mathologer shows the original proof by Euler himself. I want to thank 3blue1brown (3b1b) for hosting the Summer of Math Exposition contest (SoME1).
Time Stamps:
0:00 Intro
1:20 Act 1: Setting up the Trig Squeeze
3:09 Act 2: Putting Limits on the Squeeze
4:44 Act 3: De Moivre Becomes Imaginary
6:18 Act 4: Converting to a Polynomial
7:50 Act 5: Vieta Seals the Deal
9:34 Grand Finale and Outro
Still reading this? Then write "I love the Basel Problem!" in the comments below!
#3b1b #SoME1 #risetotheequation
DISCLAIMER: Links in this video description might be affiliate links. If you purchase a product or service using one of these links, I may receive a small commission at no additional cost to you. Thank you!
• Cauchy's Proof of the ...
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This is one of those super simple proofs that look sooo obvious, but the amount of insight and intuition that went into it is just staggering :)
The ease of checking the proof vs the difficulty in inventing it should be proof enough that
P =/= NP
I loved this video, except the last line, your grand finale: pi^2/6 < sum < pi^2/6.
This line can't be true. I think going to the limit makes < into
Yes, you're right! Someone else noticed the same glaring mistake. Thanks for pointing it out!
that is write, I mean right. If you right it like that, I mean, write, then it'll be false because < comparator is exclusive.
This is beautiful - well execute and presented. Curious if you know who Cauchy attacked this problem in this manner - did he come at this by this method originally or did he ultimately get to it after earlier dead ends?
@@f12mnb That's a great question! I have a feeling that after Act 2, where we see the cot^2 sum expressed (this was a we'll known departure point for the proof, but nobody had made any progress) Cauchy just started attacking it from different directions and eventually found the magical ingredients to his solution (Acts 3, 4, and 5). Of course we'll never know if he solved it overnight, a few days, weeks, or months. But knowing Cauchy I suspect it was no more than a matter of days. Remember, Newton literally solved the brachristochrone problem overnight!
Exactly what I was thinking. I was about to say the same thing before I saw your comment.
This is the proof that was included in my preparation materials for the IMO. It's much easier to follow in this video than it was on paper. Thanks!
Where can I get that material from? Please help.
I wanna say Cauchy is a genius, but that would just sound silly, it s obvious he was a one hell of a kind
He really was. I think he ranks right up there with Euler in terms of published results. Just an amazing mind.
Genius? He wasn’t merely a “genius” he was a towering intellect of staggering industry: a true Jedi-mind warrior of historical proportions festooned with legendary discoveries and insight.
Read his “sur Les integrales definees”
Crucial point of confusion in the presentation: Around @6:40, the letter "n" is being used simultaneously for two totally disconnected purposes: once as the power in de Moivre's formula (to be set to the special value n = 2N + 1) but also in its original role in defining theta_n = n*pi/(2N+1), where 1
He said "now we're starting over from a clean slate" though, so it should be well understood that it's a different n
This proof, or at least one very similar to it, appeared in a question on the admissions exam one year for the university I currently attend. They guided you through it a bit, but left some of the more interesting insights up to students to spot, so I remember it even now as a very satisfying and ingenious proof. Thank you for presenting it so clearly!
SUPER AWESOME VIDEO! Definitely a top 5 contender! Love the videos!
What a great proof! And really well explained, thank you for it.
Thank you!
br avistado
@@RisetotheEquation why in God's name would anyone EVER think of that replacement at 3:30 the noise maube but not the 2N plus 1
.thst comes out of nowhere and I don't see anyone ever doing that?? What is b N anyway? Thanks for sharing.
@@RisetotheEquation at 7:03 you realize you could also just replace the with zero to get the left hand side to equal zero..isn't that equal clever?
I love the Basel Problem!
Nice proof! I love it when limits show us the way by showing the "big picture" as smaller "finites" restrict our thinking. Oh and yes I love the Basel Problem!!
Awesome! Keep up the fantastic work!
It always starts with the circle, boys and girls. 😀
Very nice video, clear illustration of the squeeze theorem. God bless real analysis!
Excellent! Thank you for this video :)
Beautiful! I even managed to follow!
I very much enjoyed watching this video. Will add it to my "Fav. Videos" playlist!
In the beginning you assumed n
Well the solution is actually correct but the confusion is from the abuse of notation by this guy. He used n in both and sum of the cots and the derivation of the formula using de moivres theorem. In the latter just replace it with k and let k=2N+1 and it makes perfect sense.
Fantastic! Loved it
Thank you so much for this video. You have earned another subscriber. Loved your content. 😃
That was just great. Thanks 🙃 !
I have to say that it is really great. Thanks for your explanation ^^
Thank you so much!!!!!!!!!
At minute 7, you divide 0 with 0: sin(n theta)= sin ((2N+1)*(2N+1)*pi/(2N+1))=0 and sin (theta)=sin((2N+1)*pi/(2N+1))=sin(pi)=0.
Actually, all the terms in your sum are Cot(pi) which is undefined.
You calculated both the numerator and denominator of the equality incorrectly. Look at it again, with theta=nπ/(2N+1) and n=2n+1. Sin^n(theta)≠0 with the substitution above.
Yes, you must be right. At my first watching, I began losing the thread of the explaination at 4:50. The reason seems to be the mess with using small n and capital N, starting from that point. It requires more detailed derivation. If one replaced accurately some instances of n with N and, then, introduced another auxillary variable, say k, for summation, they may well get the correct intermediate steps.
Generally, it is a nice video! However, this mess spots the impression for those who decide to go step-by-step through it.
@@TimothyShevgun First n is the index in the sum of 1/n^2, then n is the index in the extension of sin(n theta)/sin(theta)^n, then n is the index of the rooths of the previous polynomial in n, then finallyn is back to being the index in the sum of 1/n^2. It is very confusing.
He says, "We need to carry out our substitutions for theta and n on the right hand side of the expression as well", but then doesn't do it for theta and just moves on with theta a variable, very confusing.
theta = nπ/(2N+1) = (2N+1)*π/(2N+1) = π
Indeed we have sin(nπ)/sin(π)^n = 0/0 situation.
It is better to start with dividing cos(θ)^n instead.
Now, LHS = sin(nπ)/cos(π)^n = 0/(-1)^n = 0
RHS is a function of tan((θ), but it is OK
We just solve for sum of reciprocals of roots, (1/tan(θ)^2 = cot(θ)^2),
which is the ratio of 2 coefs, on the other edge of polynomial.
Example: f(x) = (x-2)*(x-3)*(x-4) = x^3 - 9*x^2 + 26*x - 24
sum of roots = 9/1 = 9 = 2 + 3 + 4
sum of reciprocal of roots = -26/-24 = 13/12 = 1/2 + 1/3 + 1/4
Cool proof! I got a little tripped up at 4:26, since it seems to have a
Still, it should be corrected as a \leq b \leq a, since the limit doesn't preserve strict inequalities.
It should be a
@@requiemll not sure what you mean by admit doesn't preserve strict Inequalities?
@@leif1075 Well, for example, 1/n>0 (strictly!), but the limit of 1/n as n tends to infinity is equal to 0. It’s still \geq 0, but not strictly >0.
Simply Superb !
Amazing proof!!! Computing the sum with Vieta formula was absolutely brilliant! Thank you for this amazing video!
You're welcome - glad you enjoyed!!!
Great stuff!
Excellent vid, thank you so much for the time putting it together and sharing ❤
My pleasure!
This proof contains a divide by zero error and is therefore invalid. You use the substitution theta=n*pi/(2N+1) and n=2N+1, so that pi=theta and therefore sin(n*theta)=sin(n*pi)=0, to make the left side go away. However, the denominator sin^n(theta) is also equal to zero when theta=pi, so the left side of the equation becomes undefined and cannot be used. The same thing happens on the right side as cot(theta)=cos(theta)/sin(theta)=cos(pi)/sin(pi)=-1/0.
Cauchy's proof uses another variable r which takes values between 1 and N, substituting theta=r*pi/(2N+1). He then shows that the roots are tr=cot^2(r*pi/(2N+1)) which is a well defined expression.
I tripped up on this part of the proof as well, but it seems the presenter is using the letter "n" for dual purposes simultaneously, which is the source of the confusion. See my more recent comment.
Lovely!
This is amazing 😭😭
I now have a headache, but the proof was great. Still need to understand the last portions
wow amazing. Tnx!
Amazing!!!
Awesome, using only trigonometry identities and squeeze theorem 🔥
This was a nice proof!
Very elegant proof! thanks
Glad you liked it!!!
This is a super simple proof. I love it
This proof is so much in Euler’s spirit. He would have loved it!
Nice! What a whirlwind :D
I like it.
At 6:50 when you make theta=n pi / (2N+1) = pi because n = 2N+1 you seem to ignore how the denominator sin^n(theta) = sin^n(pi) is also 0. Graphing f(x)=sin(nx) / sin^n(x) near x= pi confirms my intuition that f(pi) does not equal 0, but rather diverges (to positive infinity).
If anyone can point out why the expression is instead 0 as stated in the video, please do so!
A notable and lean demonstration. There’s something to be said for expositions which use nothing more and nothing less than necessary.
Thanks. I would gladly have made it longer but the competition rules seemed to nudge the participants to keep the videos under 10 minutes.
@@RisetotheEquation It’s better to wish a song was longer than to hope it ended sooner- I believe the same principle applies here
Cool!
Very ingenious👍
He was! This guy was amazing!
i love this video :3
Thank you so much!!!
Good graphics, well presented, and well structured. But you need to issue a corrected video that has dealt with all the mathematical errors pointed out in the comments to the present version.
Thanks. I will. But in the meantime check out my Fourier proof of the Basel Problem. I think you will find it to be error-free. 😀
oh: Don't use 'n' in act 2 and in act 3. Use something else that says ' i'm an interger ' in act 3. That'll clean up your n_act3=2N+1 while n_act2=[1..N] substitution at 7:00
Yes, I was just about to say that, but you beat me to it! 😀
@@RisetotheEquation if the goal of your video was to make your audience think ... good job :-) i feel like i'm 20 again 😃
Pretty cool extension of the derivative of sin(θ).
Incredible proof
Thanks! (Really the Thanks goes to Cauchy)
Very nice. The only change I would make is to change the "
Agreed. Silly oversight on my part and I paid dearly for it in the comments.
At the end of Act 2, you've got a quantity being strictly less than itself. There is no such quantity, so either the limit doesn’t exist or those strict inequalities should’ve changed to less-than-or-equal-to somewhere along the way.
Nice one
You've immediately got a contradiction at 4:27, unless those limits do not converge. You're gonna need ≤ as opposed to < since you took a limit.
Great...but where does that sudden theta substitution come from? That should have been better stated as that is key to the whole proof.
Love the video. I am a teacher of specialist maths here in Australia. I'd love to get something that transitions equations like that. Do you have any tips?
Yes. I use PowerPoint. It has a nifty feature called the morph transition which does the equation animation. I may do a tutorial video on that very soon, so stay tuned!
Very nice.
PS: As I understand it, Euler's proof was not rigorous according to today's standards, because he was a bit sloppy with limits.
Correct. Though I wouldn't have known the difference. If Euler says it's true, I am not going to doubt it. 😀
The reason was he didn t have a proof for his product formula for the sine function. But a few years later, in the 1740s, he gave another proof that didn t depend on it.
@@raphaelreichmannrolim25 Thanks for info. Euler had some incomplete proofs during his career, for example, the case p=3 in Fermat's Last Theorem
Nice proof
Very nice. Does anyone have a reference to Vieta's theorem?
Should have thought. Fundamental theorem of algebra, expand to linear in x. Fundamental theorem of algebra pretty easy using complex variables.
Clean proof
Best one
thats was a genius solution wow
Pretty cool
Wow!
at t=6:58, Theta=Pi, but sin(n*theta)/(sin(theta))^n is becoming 0/0 form hence it will not be equal to zero, you have to find the limit. I think so.
little error at 4:15, when applying the limit to the inequality your < become ≤. the proof is straightforward. also it‘s obvious later when you have got the situation π²/6 < x < π²/6. that‘s obviously not true since e.g. strict orderings are antireflexive
Thanks! I'm keeping track of these little errors and will pin an errata soon, so I appreciate your comment!
But 6(or any number) is neither less than 6 nor greater than 6. Which means 6=6. What's the problem with that. Didn't understand what you said but sorry I don't know much about maths indepth. So please enlighten.
Although Cauchy was also a giant, it was the great Euler who first discovered it.
Indeed - one of many great discoveries that made him famous!
I didn't get the sin(n theta)/sin^n(theta) = 0. If you do your substitution you get sin(n*pi)/sin(pi) soo I'm a bit confused looks like 0/0 to me
First inequality also includes theta=0 case because theta is arbitrary so inequality becomes less than or equal to
So, pi^2/6 < ... < pi^2/6? (/raises eyebrow).
Other than that small typo, this was an awesome video!
Thanks, but nobody will ever forgive me for that titanic error !
Euler's proof IM (one cent) O is simpler and elegant just like how he proved Leibniz's series.
that's a nice proof
where did the npi/2N+1 come from please explain i dint understand
Please search the comments and you will find some good explanations (I know this one of the trickier parts of the proof).
I love the epic music at the end hahahahaha
Lol...glad you like it!
Like that " rise to the equation ' 😂
As was already pointed out, < must be replaced by
Thanks for pointing it out. Silly mistake, but hopefully the overall structure and idea of the proof makes good sense.
@@RisetotheEquation Of course, it was a very interesting and informative video. Thank you and keep it up!
As far as I know this proof is due to Ivan Niven. It is given in the appendix of his famous number theory book...
10/28/21: this channel has 3.14k subs. Very fitting
Astute observation!
Later that day….3.17…. a fleeting metaphor
you lost me at 7:00. it seems to me that sin(n.pi)=0 all right but also the denominator sin^n (pi). so it is not 0 but 0/0 and actually lim (x->pi) of sin(nx)/sin^n(x) is infinte for any n larger than 1.
An all the cotangents are undefined too. The whole equality is wrong.
Just an application of L'Hôpital's rule?
No no just horrible notation. Use another variable k in the formula of sin(k.theta) /sin^k(theta) . Now let k=2N+1 and theta is all right.
I don't understand the equality: Angle theta = n*pi/(2N + 1), I need an explanation. THANKS
please just move the whole ecuation, or morph it completely before moving it, but for the love of god if you're not applying distributive properties DO NOT use the individual character mixup to make transitions, it's confusing as hell.
Thanks! I'll be sure to keep that in mind next time.
nice
2:36 min Theta should be in radians for this formula to work. Units are important!
Doesn’t the step at 7:00 lead to a 0/0 error?
Math is so beautiful 😍. Too bad people can't appreciate its beauty. For its useful purposes. Here is something that you don't see often unless you are in a trade. An=5*92 [36-n/39]^2
The numbers precede forms. The lie in the middle between the essence and the action (manifestation) of the divine !!!
Fourier series method is my favorite
Fourier proof is awesome, I agree!
There is an error on the second to last slide -- less than (
Yes, thanks for catching this glaring error!
Most of the video was very clear, and I really like the clean visuals.
While the steps are easy to follow due to your clean presentation style, it’s not clear where any of this is going until 3:43 when the middle value becomes the series we are interested in. This has two issues:
While I have seen 3b1b’s video, the fact that 1/n^2 = 1/1+1/2^2+… is not stated explicitly in your video, and is only hinted at by your title card. Seeing as this is the core of the video, mentioning it would have been a good move.
Secondly, it is nice to know where a trail is meant to lead before we are at its end. You draw a circle, some triangles, find their area, write an inequality, and multiply said inequality by seemly random values before we reach the equation we are interested in. It’s clear in the end what it was building towards, but it would be helpful if the sections of the video progressed with a goal in mind instead of retroactively pointing out the path we were on the whole time.
I appreciate that when you replace a part of an equation with its equivalent, you have both on screen for a few seconds for us to verify why the two are equivalent.
I think this is an artifact of the software you are using, but at 4:12 (and several other times), the 3 equations break apart and reform out of pieces they were not initially made of (bits of the right equation end up in both the top and bottom equations.) The animation looks like a lot is going on and I got a little confused. By flipping +/-5seconds with the arrow keys could I confirm the only change between the 4:09 and 4:14 was the addition of limit bars. This happens a few times where 2 or 3 equations need to move around the screen, but instead of moving as one piece, it turns into a shell game.
It may have been helpful to state why you can break up the real and imaginary parts of an equation and equate them at 6:06. If you briefly explained why this works (you can’t rotate into or out of the imaginary axis by adding) the amount a viewer would need to already know about the complex plane to understand the video would decrease significantly.
At 7:22, you write t=cot^2o. The rest of the video, your substitutions are generally replacing the left item with the right, which had me wondering how I missed t this whole time. I know directionality is entirely aesthetic, and most of the time the singular variable is on the left, but since you are using this equation as an operation on another equation, it might be better to write it in a way that better suggests its function.
Overall, I think you made a pretty good video. Sorry I took so long to notice you comment asking for feedback, and thank you for making the outro music so quiet.
Thanks for the feedback! The glitch you're referring to at 4:12 and other parts of the video isn't a glitch, actually. It's the PowerPoint morph transition, so it's morphing the characters of the equation from the equations on one slide to the equations on the next slide. So, yes, you're seeing little bits of the equation moving around. I may speed up the transition in future videos.
@@cfgauss71 Oh, I didn't realize PowerPoint was now capable of animations like this and assumed you were using something more complicated. Speeding up the animation would stop me from thinking the bits of equation were getting rearranged when they weren't, but if you move the equations, then alter them in a 2 step process, I think it would make things more clear. It's comprehendible as is, I just think it could be clearer.
Well said！
I think that the end not looking correct as PI/6 < Sum < Pi/6 should be in fact PI/6 =< Sum =< PI/6 ???
Yes, that's right. But in fact, the proof is correct because you have to replace < with
Thanks for spotting this tiny yet glaring error!
@@RisetotheEquation NP I am proud of having noticed this as I only understood half of theses HL maths, but I am working on it ;-) TY for your work.
Literally used most branches of mathematics to come up with this. How on earth did he come up with the substitution theta = (npi() / 2N + 1)?
I am a little confused how n can simultaneously be less than N and yet = 2N + 1. Can someone help me understand?
Well the solution is actually correct but the confusion is from the abuse of notation by this guy. He used n in both and sum of the cots and the derivation of the formula using de moivres theorem. In the latter just replace it with k and let k=2N+1 and it makes perfect sense.Hope it helps.
Man That was fast
bprp mentioned = like in the video and sub in the channel : )
I still don't understand why you can substitute n=2N+1
Nice they made this into a step question lol
Can anyone clear up how theta=npi/(2N+1) at 3:31 is derived? What if theta was e/√7?
I think you wantend him to show that it is possible to rewrite any θ as nπ/(2N+1), but that's indeed wrong as you pointed out (you can't get θ=e/√7 ). However that's not what he's saying. He's saying that, since that inequality is valid for any θ between 0 and 90°, it must as well be valid for θ=nπ/(2N+1), as long as it's between zero and π/2, and it is:
nπ/(2N+1)2n since he stated 1
@@Fin8192 Thanks
@@Fin8192 Nice explanation just a little nitpick. I'm pretty sure no one can prove that theta can't be e/√7 . As that would be very close to proving that e/π√7 is irrational.
The proof of the rule at 8:00 is so easy, I don't know why you didn't go over it?
Because it's so easy?
Namaste.
The area of triangle OAB is also a textbook formula (depends on the textbook of course). We were taught that if you have 2 sides of a triangle a and b and the angle C between them the area is 1/2 ab sin C
Appreciate the mini-proof of that formula though.
The equation would be more beautiful if you moved the 6 to the numerator on the right, and represented it as 3!
Seriously... I didn't know that high schoolers somewhere sre taught De Moives theorem
In the UK, De Moivre's theorem would be covered in A level Further Maths, when students are 17-18 typically.
I just think that it's even sweeter than Euler's method.
I mean, how on earth can you see this problem and say to yourself,
"Duh! Let's just assume a circle and inscribe a triangle. "??!!?
I agree!!! Thanks for watching! 😀
this proof was first used to proof the basic limit sinx/x as x->0, from the inequality sinx
maybe, knowing that the sum was π²/6 prompted Cauchy to view the terms as angles, etc?