@@Paul93Ye It definitely helps. I did well in math, but of course cannot compete with these guys. If somebody just gave me the number 1.644934 and asked what it was approximating, nothing in particular would come to mind. Maybe after a while I could figure it out, but I'm sure no competitor to Euler.
How would you get 16 digits without a computer? You couldn't just add up terms, because it doesn't converge quickly enough. It would take over 100 million terms. Could it perhaps be done with calculus, finding the area of the smooth curve then estimating the error? I would need to see the method before being confident he actually had 16 digits.
@@jumbojimbo706 it’s not showing off it’s a standard thing when you learn the Fourier series. We did the exact same thing plus a few other nice identities for homework
Omg I recently proved a result on my example sheet on Fourier Series by accident to show the sum of the reciprocal fourth powers is pi^4/90, I believe if u take the FS of (1-x^2)^2 periodic on (-1,1) gives u this result. I was so fascinated by those, I did more research and managed to find a beautiful closed form solution for zeta(2n), so for n=2,4,6,…
But then your friend isnt doing harder problems than euler, because relatively euler was doing the most difficult problems of his time and even far ahead of them
My favorite proof of this is to look at a Fourier series of a sawtooth signal, say y = x from -pi to pi. Each sine term will have a 1/n factor. The power of the signal is just the integral of its square, so you end up getting the squares of all the individual sin terms in the series. Equating both sides (the original signal, and it's series) results in sum(1/n^2) on one side, and pi^2/6 on the other. I found this on my own and then learned it was well-known.
That's cool, very impressive. I derived the focal length of a spherical section mirror when I was 16, and took the limit as the size gets small, and showed it around my high school. I couldn't get people to believe I didn't find it in some old book. (That was before the internet existed.) It took all the math I had taken up to then, which did not include calc.
@@yahyabenhissoune Thanks. I'm not sure, I just moved and I probably can't find my old papers from high school but I'll look. That was 1986, time flies and details of memories fade, but the general idea was that I calculated the angle of reflection, then took the limit as the size of the mirror approached zero.
Great explanation of this solution. 👍 I think my favorite video on the Basel Problem is the one 3Blue1Brown did where he showed geometrically where the “hidden circle” is in the equation by calculating the brightness of lanterns around a large circle and then looking at the limit as the circle’s radius approaches infinity.
@@andreasxfjd4141 Regardless... I'm sure Euler based his deductions on logical reasoning. It makes a lot of sense that a polynomial can be written as products of its roots, so he went with that. And on and on... until he arrived to a result, that when calculated, yields the exact values that you would get from brute force calculating 1 + 1/2! + 1/3! + ... Either it was a huge coincidence, or he stumbled on an important mathematical result
@@andreasxfjd4141 The result is not based on the fundamental theorem of algebra, but on the Weierstrass factorization theorem, which extends the fundamental theorem of algebra from the integral domain of polynomial functions to the integral domain of entire functions.
As a student at Technical University - I had A grades from calculus and B+ from linear Algebra. I am also Math passionate from primary school, but my skills were boosted by very particular and demanding teacher in secondary school.You use a lot of tricks which are known for me, but some are really brillant and I see few of them first time! To proof equation mentioned in this video I use a bit different approach. I use Fourier series. To be more meaningfull - I use express function f(x) = ×^2 for x and then with help Dirichlet's conditions and with Fourier's series we can obtain pi^2/6. I am amazed with you calculus skills. Quite decent! You are doing it really great! When I see it, I am a bit claimer, that there are on earth Math passionates like me ;). Cheers!
Wow, I am loving your videos! Sometimes people just launch into an explanation without taking a step back and giving a broader context, or discussing their approach. I really appreciate how you structure your videos, and how you explain the concepts inside of them. Keep up the great work!
For those who are curious about the manipulation that factored sin(x)/x into infinitely many monomials, this is made rigorous by the Weierstrass factorization theorem, which is a generalization of the fundamental theorem of algebra.
@@bakhridinova6482 it's quite a nuanced topic to cover in Bri's short video style, I instead recommend Mathologer's video on it: th-cam.com/video/YuIIjLr6vUA/w-d-xo.html
Hard enough to solve the damn problem, let alone Edit in all the graphics, key frames, lighting effects, and scripting to make it as simple to understand, Good job man! :D
Great video. A minor (yet important) correction : in timestamp 1:56 , in the last line, the "greenish" zero should be inside curly braces, since "backslash" subtraction is defined between two sets ( it is undefined between a set and an element.)
U should do a video extending this formula to zeta(2n), I would love that! Recently did a talk on it using a very similar method which uses the weierstrass infinite product for sinh rather than sine, to generalise it for n = 2,4,6,…
the taylor series is the difficult part. The rest is basic I would say. The more impressive is how he connected the dots. And the sheer algebraic parkour he went through
If I proved that the sum of the squares of the 2 sides of a right angled triangle equaled the square of the hypotenuse in 2000 BC, I'd be world famous.
Calculating (-1/2)! by a method adopted by myself - Let's calculate C(n,1), of course it is n . Put n=1/2 so C(1/2,1) is equal to 1/2 apply formula of combination C(1/2,1)= (1/2)!/{1!(-1/2)!} . Now knowing 1/2! as √π/2 , equate both equations and hence we get value of (-1/2)! as √π . Incredible . Similarly we can calculate some more negative and fractional factorials . If you know this trick already, then this trick has been already discovered, but if no one knows this trick then I am the first to use this .
I have a question. If you had the product of a Power Series’ roots, like we did in this problem (1+x/π)(1-x/π)••• How can you then find the power series for that? In other words, the product of the roots in this video also happens to be the product of the roots for Tan(x)/x so how do we know the power series isn’t this, but it’s sinx/x
Well not everyone likes Tiktolk I personally cring when watching it. Once I cringed so hard to the point where I just couldn't watch people try to act/be funny, and we all know that people mostly watch Tiktok to laugh/disconnect from reality.
@@Reports. Just because you and some others (including myself) do not watch the tiktoks doesn't mean that it is not a way to fame and some great money, though!
@@grubbygeorge2117 I never claimed that it is not, I'm just mentioning that it is not something everyone likes because of all the simps, attention seekers, psychos, and ''girl power movement/man-hating''. Given there are actually funny people there but the majority are the states above.
@BriTheMathGuy I tried something similar with exp(x) - 1 instead of sin(x) and got a different result. Is there a reason why? Did I do math incorrectly?
We just are considering the terms with x^2 in them after distributing out the whole thing. Just look at the first few: -x^2/pi^2 *1, we’re multiplying by 1 for all of those terms so all the terms with x^2 in them stay the same and are added up to get the final coefficient for x^2, then we just factor out x^2
@@wavez4224 well, we just took this representation of (sin) and (cos) and (e) to know how did Euler’s formula come, so we don’t actually use those representations in our questions
@@wavez4224 but yea overall i’m having fun with my math study of high school :) most ppl here don’t because they only rely on what we learn from school or whatever, which is mostly boring and not taught in the best way, but i dig inside the meanings of what i learn throw youtube and have a deeper perspective about it
@@Z7youtube yea I also felt the same in high school. It felt like we weren’t being taught the full version because most people only take the course to get the credit. I’m now studying computer science in university but considering a switch to math major I’m guessing you already know about him but 3blue1brown posts phenomenal videos about math topics. You should check out his videos if you haven’t already.
It is kinda annoying and highly nontrivial, though, to justify the factorization which was proven much later and is known as Weierstrasse factorization theorem.
To solve this strictly without proofing, that INFINITE polynomial may be factorizated, we can consider the function (sin πx)/πx In the first of all, let's express this function in the Maclaurin series: (sin πx)/πx=1-π²x²/3!+π⁴x⁴/5!... On other side, we have Fourier series of piecewise-defined function f(x)=cos αx, where x∈[-π;π] and α∉Z: cos αx=((sin πα)/π)(1/α+∑{n=1, to ∞}((-1)ⁿ(2α/(α²-n²))cos nx)) Put α=t and x=π. Divide by (sin πα)/π and... We get inf. sum for π cot πt into aliquot fractions: π cot πt=1/t+∑{n=1, to ∞}((-1)ⁿ(2t/(α²-n²))), where t∉Z Subtract from the both parts of equality 1/t, integrate it from 0 to x and use one of rules of log (if logarithms equal, their arguments equal too), we get infinite multiplication for (sin πx)/πx: (sin πx)/πx=∏{n=1, to ∞}(1-x²/n²)=(1-x²)(1-x²/4)(1-x²/9)(1-x²/16)..., x∈(-1;1) (omg, I'm tired) From equal of Maclaurin and Fourier series we have: 1-π²x²/3!+π⁴x⁴/5!-...=(1-x²)(1-x²/4)(1-x²/9)(1-x²/16)... -x²(π²/6)=-x²ζ(2), ζ(x) - dzeta-function. (In 2 it equale infinite sum of n¯². And that's what we need) Divide by -x² and finally get: ζ(2)=π²/6 I found this beautiful modification of First Euler's method of solving Basel's Problem absolutely accidentally, when I decided to integrate infinite sum expression of π cot πt to practice XD I'm sorry for grammar and lexical mistakes, if I did them. I don't speak English very well :c
@@angelmendez-rivera351, yes... But I wrote "...without proofing, that INFINITE polynomial may be factorizated...". Factorization of infinite polynomials isn't obvious. I proofed calculation with a little other idea)
You only use the location of the roots of the function and the fact that the zeroth order of the Taylor series is 1. What would happen if you would take another function with these properties,for example (0.7*sin(x)+0.1*sin(3x))/x? This is another function with a different Taylor series, but the product representation with the roots is the same. Isn't this a contradiction?
This is beautiful but how do we know that the rules which hold for polynomials with finite number of terms will also hold for an infinite polynomial (or a power series whatever you call it? ) Like matching terms with same power of x.
Matching terms is correct for power series. You could see a powerseries as a vector in the infinite dimensional vector space P(R). Since it is an vector, it has a unique representation in terms of the basis, which are all positive integer powers of x. Therefore two representations of the same vector (the power series) must have the same coefficients.
@@thijsdekok798 What you said is not quite correct, and it is a well-known fact that this factorization only works if the series has an infinite radius of convergence, which it does in the case of sinc(x).
🎓Become a Math Master With My Intro To Proofs Course! (FREE ON TH-cam)
th-cam.com/video/3czgfHULZCs/w-d-xo.html
❤
If Euler proved something in 2021 he'd be even more famous.
For a number of reasons in fact.
@@glumbortango7182 including how he is still alive in 2021 to prove something
@@dionysianapollomarx
My point.
Imagine Euler made some secret proof but hid it an it was only found hundreds of years later
If Gauss or Euler still lived today, the history would be different
Cool fact: Euler actually approximated the sum to 16 decimal places and GUESSED that it was pi^2/6 before rigorously proving it
Woah! I did not know that, cool!
16 digits is a whopping good approximation.
I think if you know the result, how to prove it becomes easier with backwards engineering. Still impresive.
@@Paul93Ye It definitely helps. I did well in math, but of course cannot compete with these guys. If somebody just gave me the number 1.644934 and asked what it was approximating, nothing in particular would come to mind. Maybe after a while I could figure it out, but I'm sure no competitor to Euler.
How would you get 16 digits without a computer? You couldn't just add up terms, because it doesn't converge quickly enough. It would take over 100 million terms. Could it perhaps be done with calculus, finding the area of the smooth curve then estimating the error? I would need to see the method before being confident he actually had 16 digits.
I proved this result using a Fourier series as one of my homework assignments for my math class but the way Euler did it is extremely elegant.
Parseval’s identity?
I did it slightly differently. I expanded x^2 in terms of an exponential Fourier series and evaluated the series at the boundary (x=1).
Ok lad it’s youtube comments you ain’t gotta show off
@@jumbojimbo706 it’s not showing off it’s a standard thing when you learn the Fourier series. We did the exact same thing plus a few other nice identities for homework
Omg I recently proved a result on my example sheet on Fourier Series by accident to show the sum of the reciprocal fourth powers is pi^4/90, I believe if u take the FS of (1-x^2)^2 periodic on (-1,1) gives u this result. I was so fascinated by those, I did more research and managed to find a beautiful closed form solution for zeta(2n), so for n=2,4,6,…
After watching this, I understand why the people of 1734 would make you famous. That was some serious deduction.
Ya gotta hand it to Euler!
@@BriTheMathGuy he had a good eye for math, he really did!
@@peterfireflylund too bad he lost it
But then your friend isnt doing harder problems than euler, because relatively euler was doing the most difficult problems of his time and even far ahead of them
@Good Game i'm hopefully wanna see some tremendous achivement of your friend in the future
Wow, super well done man! This isn't just math anymore, it's mathemagic.
Very glad you enjoyed it!
At some point your channel is gonna be Big! You make so easy explanations for difficult problems, and this is awesome!
Pero bueno maik, que haces acá también jajajaja
I appreciate that! Thank you very much for watching and have a great day!!
@@Kevin-14 estoy en todas partes jeje
Maths Mike studying its competence. 😂 😂 😂
La verdad, tras ver el video he pensado que esto es algo que encajaría en tu canal. ❤️
I already recommended it for my kids . A talented teacher.
My favorite proof of this is to look at a Fourier series of a sawtooth signal, say y = x from -pi to pi. Each sine term will have a 1/n factor. The power of the signal is just the integral of its square, so you end up getting the squares of all the individual sin terms in the series. Equating both sides (the original signal, and it's series) results in sum(1/n^2) on one side, and pi^2/6 on the other. I found this on my own and then learned it was well-known.
That's cool, very impressive. I derived the focal length of a spherical section mirror when I was 16, and took the limit as the size gets small, and showed it around my high school. I couldn't get people to believe I didn't find it in some old book. (That was before the internet existed.) It took all the math I had taken up to then, which did not include calc.
@@yahyabenhissoune Thanks. I'm not sure, I just moved and I probably can't find my old papers from high school but I'll look. That was 1986, time flies and details of memories fade, but the general idea was that I calculated the angle of reflection, then took the limit as the size of the mirror approached zero.
@@karldavis7392 wow if i discovered that i'll proudly flaunt around with that result too
Great explanation of this solution. 👍 I think my favorite video on the Basel Problem is the one 3Blue1Brown did where he showed geometrically where the “hidden circle” is in the equation by calculating the brightness of lanterns around a large circle and then looking at the limit as the circle’s radius approaches infinity.
Yes, that's a great follow up to this video! Here's the link.
th-cam.com/video/d-o3eB9sfls/w-d-xo.html
I really enjoyed this derivation of the famous sum. The application of the fundamental theorem of algebra to solve this is genius.
As this sum was solved (calculated), the fundamental theorem of algebra was not proved.
@@andreasxfjd4141 Regardless... I'm sure Euler based his deductions on logical reasoning. It makes a lot of sense that a polynomial can be written as products of its roots, so he went with that. And on and on... until he arrived to a result, that when calculated, yields the exact values that you would get from brute force calculating 1 + 1/2! + 1/3! + ...
Either it was a huge coincidence, or he stumbled on an important mathematical result
@@andreasxfjd4141 The result is not based on the fundamental theorem of algebra, but on the Weierstrass factorization theorem, which extends the fundamental theorem of algebra from the integral domain of polynomial functions to the integral domain of entire functions.
Euler was indeed a genius!
You explained this so easily wow .. Thank you so much. 💕
You're so welcome!
As a student at Technical University - I had A grades from calculus and B+ from linear Algebra. I am also Math passionate from primary school, but my skills were boosted by very particular and demanding teacher in secondary school.You use a lot of tricks which are known for me, but some are really brillant and I see few of them first time! To proof equation mentioned in this video I use a bit different approach. I use Fourier series. To be more meaningfull - I use express function f(x) = ×^2 for x and then with help Dirichlet's conditions and with Fourier's series we can obtain pi^2/6. I am amazed with you calculus skills. Quite decent! You are doing it really great! When I see it, I am a bit claimer, that there are on earth Math passionates like me ;). Cheers!
Wow, I am loving your videos! Sometimes people just launch into an explanation without taking a step back and giving a broader context, or discussing their approach. I really appreciate how you structure your videos, and how you explain the concepts inside of them. Keep up the great work!
Thanks so much!
Your explanation is the best I've seen on ytube; concise and clear.
Glad you liked it!
For those who are curious about the manipulation that factored sin(x)/x into infinitely many monomials, this is made rigorous by the Weierstrass factorization theorem, which is a generalization of the fundamental theorem of algebra.
You are already famous! 58k subs isn't a joke :)
The title says world famous
@@speedlunary797 Its not that people are watching him from only one country... People from many countries are watching him
Makes me wonder. In his lifetime did 58k people read Euler?
What Famous result should we look at next?!
1+2+3+...=-1/12 please
gaussian integral
@@bakhridinova6482 it's quite a nuanced topic to cover in Bri's short video style, I instead recommend Mathologer's video on it: th-cam.com/video/YuIIjLr6vUA/w-d-xo.html
@@danielmago4327 already done th-cam.com/video/S79KPrIm_Gc/w-d-xo.html
integral of sin(x)/x
This was a really well made video.
Glad you thought so! Thanks for watching!
And by curiosity i found that- Summation of 1/{(2n+1)^2} from n=0 to n=∞ is => π^2/8.
Love your videos!
Thank you! Have a great day!
I love this! Thank you!
You're so welcome!
Amazing video, as always!
You're the best!
Hard enough to solve the damn problem, let alone Edit in all the graphics, key frames, lighting effects, and scripting to make it as simple to understand,
Good job man! :D
How man?? How with such ease?? Hats off 🙏♥
Thank you! Cheers!
Loved the video and the development. 👌
Glad you enjoyed it!
Great video. A minor (yet important) correction : in timestamp 1:56 , in the last line, the "greenish" zero should be inside curly braces, since "backslash" subtraction is defined between two sets ( it is undefined between a set and an element.)
You're knowledge! Hats off 💫👑
excellent explanation. thank you
You deserve to be famous as you managed to explain this in 4 minutes
Yeah I totally understood all that.
WOW, I didn't understand most of it but it seem that for someone with math knowledge bigger than first semester you will teach him great
Glad to hear that! Thanks for watching!
And I was sitting here 95% of the time thinking the answer must be 2...
And there are still people out there saying Euler isn't a physicist
U should do a video extending this formula to zeta(2n), I would love that! Recently did a talk on it using a very similar method which uses the weierstrass infinite product for sinh rather than sine, to generalise it for n = 2,4,6,…
We were TRICKED into proving this in Signal processing class (in a slightly backwards way of course)
Beautiful!
Yea I’m in algebra 2 and trigonometry I have no clue what’s going on right now
the taylor series is the difficult part. The rest is basic I would say. The more impressive is how he connected the dots. And the sheer algebraic parkour he went through
this channel is awesome
Glad you think so! Have a great day!
If I proved that the sum of the squares of the 2 sides of a right angled triangle equaled the square of the hypotenuse in 2000 BC, I'd be world famous.
haha.
If I was born in 100000 BC I'd be world famous
3:40 Sum body once told me...
😂
that is ABSOLUTELY brilliant!!!!!!!!!!!!!!
Awesome solution man.
Glad you think so!
Class 11th student watching this❤
Me at night watching this before I go to bed:
"Yeah I think I get it"
Petition to send this back in time.
“Read Euler, read Euler, he is the master of us all.” -- Laplace
This video just brought joy to me.
Very happy to hear it! Have a wonderful day!
Majestic Euler clicked it 🗿
Calculating (-1/2)! by a method adopted by myself -
Let's calculate C(n,1), of course it is n . Put n=1/2 so C(1/2,1) is equal to 1/2
apply formula of combination C(1/2,1)= (1/2)!/{1!(-1/2)!} . Now knowing 1/2! as √π/2 , equate both equations and hence we get value of (-1/2)! as √π . Incredible .
Similarly we can calculate some more negative and fractional factorials .
If you know this trick already, then this trick has been already discovered, but if no one knows this trick then I am the first to use this .
Wow just brilliant 😍
Glad you like it!
I didn't understand most of this and it make want to learn more of It.
The volume of a 4dimensional sphere with radius 1 is: π²/2.
So the basel problem is a third of that. Maybe you could prove it with 4-spheres.
Interesting
That’s great video
When i was using a n infinite series calculator, i accidently put in this and i remembered what 3b1b said .
I have a question. If you had the product of a Power Series’ roots, like we did in this problem (1+x/π)(1-x/π)•••
How can you then find the power series for that?
In other words, the product of the roots in this video also happens to be the product of the roots for Tan(x)/x so how do we know the power series isn’t this, but it’s sinx/x
I have a doubt if we write in the sinx/x =(1-x²/π)............
We can also write log(x) =(1-x) but reality is not correct
Brilliant!
Woah woah I just found this channel and
What do you mean such quality content isn't made by a million sub channel?
B
R
A
V
O
This is well presented for the mere mortals.
The probability that any two random numbers are Coprime is 6/pi^(2).
I love it!
To be famous in 1734:
1) know all results from high school
2) be observant
To be famous now:
1) tiktok 😑
😂
Well not everyone likes Tiktolk I personally cring when watching it. Once I cringed so hard to the point where I just couldn't watch people try to act/be funny, and we all know that people mostly watch Tiktok to laugh/disconnect from reality.
@@Reports. Just because you and some others (including myself) do not watch the tiktoks doesn't mean that it is not a way to fame and some great money, though!
@@grubbygeorge2117 I never claimed that it is not, I'm just mentioning that it is not something everyone likes because of all the simps, attention seekers, psychos, and ''girl power movement/man-hating''. Given there are actually funny people there but the majority are the states above.
@@Reports. none of the things you listed are specific to the platform though
0:57 but i dont see how sinx has that product representation. i graphed it and it was not even close.
That Euler dude is pretty smart NGL
Huh...
I was thinking using the sum of squares formula for the bottom
So that.
Sum of x^2=(x+1)(2x+1)/6
And so the sum would be 1 over that. 1/S(x^2)
It didn't work since 1/(x+y) ≠ 1/x + 1/y for all reals x and y
@BriTheMathGuy I tried something similar with exp(x) - 1 instead of sin(x) and got a different result. Is there a reason why? Did I do math incorrectly?
Hlo
Solution: a time machine
Be right back! 🏃 ⏰
You should team up with Mathologer
Mathologer is my lecturer
I'd love to!
Fucking euler is in every mathematical class i take
Take a chill pill
Nice explain!
Glad you think so!
Great video :)
. i have a question, why do the x^2 gets the value x=1 at the end?
i'm a viewer from Iran
That's interesting!
Glad you thought so! Have a great day!
They are trying to take my 3rd tier proof I think I need your help
shouldn't the roots be n(pi) -1? as the constant term is 1, one of the factors may be 0 when n(pi) - 1.
(i think)
but pi² = g = 10 so pi²/6 = 1.67
Pi squared isn't equal to 10
@@jeaneude9380fundamental theorem of engineering
This guy is on the moon
Engineer moment
wonderful ❤
Thank you! Cheers!
Excuseme sir, what did you do here at 2:56 I don't understand that factorization 😢
We just are considering the terms with x^2 in them after distributing out the whole thing. Just look at the first few: -x^2/pi^2 *1, we’re multiplying by 1 for all of those terms so all the terms with x^2 in them stay the same and are added up to get the final coefficient for x^2, then we just factor out x^2
0:31 what exactly does calc 2 contain? cuz we take this representation of sin in high school
You got an interesting high school then. In my high school we only really took the derivative of functions and did optimization problems.
@@wavez4224
well, we just took this representation of (sin) and (cos) and (e) to know how did Euler’s formula come, so we don’t actually use those representations in our questions
@@wavez4224
but yea overall i’m having fun with my math study of high school :) most ppl here don’t because they only rely on what we learn from school or whatever, which is mostly boring and not taught in the best way, but i dig inside the meanings of what i learn throw youtube and have a deeper perspective about it
@@Z7youtube yea I also felt the same in high school. It felt like we weren’t being taught the full version because most people only take the course to get the credit. I’m now studying computer science in university but considering a switch to math major
I’m guessing you already know about him but 3blue1brown posts phenomenal videos about math topics. You should check out his videos if you haven’t already.
I’m in AP calc now, and I’ve gone ahead and learned stuff like this even though we’re just now mentioning derivatives in class, I wish they offered BC
1:56 sounds cut lmao
Got this video recommended and have no clue whats going on lmfaoo
It is kinda annoying and highly nontrivial, though, to justify the factorization which was proven much later and is known as Weierstrasse factorization theorem.
Well, you are famous even now.
Really enjoyed this! We can do this sort of infinite-term manipulation provided that the series converges absolutely, correct?
Yes.
Absolute convergence is not sufficient, actually. You also need the series to have an infinite radius of convergence.
Thought he would do sum of 2^n equals -1 😁
I will challenge you to make a 10 minute video on fermat's last theorem where you will show the proof.
I would fail that challenge!!
Real challenge: be world famous in 2021
😬
You can still be world famous, if you prove the P/NP conjecture.
I'll get right on it 🤔
To solve this strictly without proofing, that INFINITE polynomial may be factorizated, we can consider the function (sin πx)/πx
In the first of all, let's express this function in the Maclaurin series:
(sin πx)/πx=1-π²x²/3!+π⁴x⁴/5!...
On other side, we have Fourier series of piecewise-defined function f(x)=cos αx, where x∈[-π;π] and α∉Z:
cos αx=((sin πα)/π)(1/α+∑{n=1, to ∞}((-1)ⁿ(2α/(α²-n²))cos nx))
Put α=t and x=π. Divide by (sin πα)/π and... We get inf. sum for π cot πt into aliquot fractions:
π cot πt=1/t+∑{n=1, to ∞}((-1)ⁿ(2t/(α²-n²))), where t∉Z
Subtract from the both parts of equality 1/t, integrate it from 0 to x and use one of rules of log (if logarithms equal, their arguments equal too), we get infinite multiplication for (sin πx)/πx:
(sin πx)/πx=∏{n=1, to ∞}(1-x²/n²)=(1-x²)(1-x²/4)(1-x²/9)(1-x²/16)..., x∈(-1;1)
(omg, I'm tired)
From equal of Maclaurin and Fourier series we have:
1-π²x²/3!+π⁴x⁴/5!-...=(1-x²)(1-x²/4)(1-x²/9)(1-x²/16)...
-x²(π²/6)=-x²ζ(2), ζ(x) - dzeta-function. (In 2 it equale infinite sum of n¯². And that's what we need)
Divide by -x² and finally get:
ζ(2)=π²/6
I found this beautiful modification of First Euler's method of solving Basel's Problem absolutely accidentally, when I decided to integrate infinite sum expression of π cot πt to practice XD
I'm sorry for grammar and lexical mistakes, if I did them. I don't speak English very well :c
By the way, we can very easy solve Basel Problem by Fourier series of x²))
You said this is a strict calculation without "proofing" (?), but what you wrote is an alternative proof anyway
@@angelmendez-rivera351, yes... But I wrote "...without proofing, that INFINITE polynomial may be factorizated...". Factorization of infinite polynomials isn't obvious. I proofed calculation with a little other idea)
@@angelmendez-rivera351 I think, that I just did grammar mistake in that sentence) Isn't it?
Bro I did this when I was 9 when I was dreaming of infinity
If I had done it in 1734 I'd have been famous. Grammar police!
Being famous by telling others idea sucks, it felt like doing evil thing
😈
@@BriTheMathGuy hahaha im sorry, i dont mean to give hate messages or something like that. Im just being sarcastic
@@mfadhilal-fatih1427 not at all! Have a great day!
Amazing
You are!
my head hurts now
I'm sorry but I hope you have a great day anyhow!
@@BriTheMathGuy thank you
also can you too a video on what it means to take a number say x to the hyperpower of a fraction I've been curious
@@Logicallymath I don't know the answer currently but that sounds interesting!
@@BriTheMathGuy Even Bri the math guy doesn't know! OH NO thanks for your time
You only use the location of the roots of the function and the fact that the zeroth order of the Taylor series is 1. What would happen if you would take another function with these properties,for example (0.7*sin(x)+0.1*sin(3x))/x? This is another function with a different Taylor series, but the product representation with the roots is the same. Isn't this a contradiction?
I hope he will cross 500k by end of this year
😲
This is beautiful but how do we know that the rules which hold for polynomials with finite number of terms will also hold for an infinite polynomial (or a power series whatever you call it? ) Like matching terms with same power of x.
Matching terms is correct for power series. You could see a powerseries as a vector in the infinite dimensional vector space P(R). Since it is an vector, it has a unique representation in terms of the basis, which are all positive integer powers of x. Therefore two representations of the same vector (the power series) must have the same coefficients.
@@thijsdekok798 What you said is not quite correct, and it is a well-known fact that this factorization only works if the series has an infinite radius of convergence, which it does in the case of sinc(x).
We know these rules are valid thanks to the Weierstrass factorization theorem.
Angel Mendez-Rivera I know, but I wasn’t talking about the factorization, solely about the matching coefficients
I KNEW IT!
Great! Have a wonderful day!
oiler 💀
"oiler"
Wow!
🤯