Proof Of The Basel Problem

First conceived by Pietro Mengoli in 1650, Leonhard Euler proved this seemingly impossibly infinite series in 1734. The problem is named after Basel, the town in which Euler was born.

[ikarienator]

The Weierstrass expansion needs to be explained better. Otherwise sin(x) = 2sin(x) = sin(x)^30 = exp(x) sin(x) because their roots coincide.

[geenstagni]

Beautifully explained! I'm an American uni student that has probably bit off more than I can chew in terms of my Real Analysis class so I turned to your video for help and I cannot thank you enough for taking the time to explain each and every step from scratch. Wikipedia, Math exchange, lecture notes, etc. have all failed me due to omitted "known" information but you sir, have not.

Admittedly, I was a little confused at the part in which you compared the coefficients of both sequences, but I eventually got it. Your video put a smile on my face by making me understand maths again! Thanks you!!

[IsnardLopes]

Thumbnail said "Proven carefully", the first opperation the dude does is to divide by x, without assuming that x is not 0. (Contains irony xD)

[krabkrabkrab]

It's a beautiful proof, mostly well-stated. The missing piece is that one has to impose that sin(x)/x =1 at x=0. That is the reason it's divided through by n^2pi^2. For example, a polynomial p(x)=2x^2-8x+6 has zeros 1 and 3. But then it is incorrect to assume p(x)=(x-1)(x-3). Knowing p(0)=6 demands the (x-1)(x-3) be scaled up by a factor of 2.

[dr.osborne]

Not rigorous without a proof of the Weierstrass factorization thm. I think this is the reson why Euler's contemporaries did not accept his first proof. Weierstrass came much later. How do we know, for example, that the product you reference does not converge to e^(-x^2) times sin(x)/x ? I do the same thing in a math class I teach, but always am concerned because of this missing piece. That class doesn't have the analytic function theory needed for the complete proof and I thought this video would show me a different way.

[elenaasi]

That was amazing, sir.Thank you for your time and effort.I don't usually comment under videos, but yours was really useful.You just gained a new subscriber.

[realcirno]

fourier series direct method: the fourier series of |x| on [-pi, pi] is pi/2 - (4/pi) sum(n odd >= 1) cos(nx) / n^2, this converges absolutely to |x|, if you let x=0 and do some algebra you get basel sum (its actually not too difficult to show that the fourier series converges uniformly to the original function if the sum of the coefficients converges absolutely, i would argue the analytical machinery behind that is much simpler than whats needed for weierstrass product)

residue theorem method: the function f(z)=cot(pi z)/z^2 has a triple pole at 0 and a simple pole at other integers, the residue at 0 is -pi/3, the residue at n not equal to 0 is 1/(n^2 pi). it can be shown with a square contour that if g(z) is O(1/z^2), then the sum of the residues of g(z)cot(pi z) is equal to 0, hence -pi/3 + (2/pi) * sum from n=1 to infty of 1/n^2 = 0. (this method generalizes very neatly for calculating many different infinite series of the form 1/P(n) where deg P >= 2)

[小達-cd]

6:41 you have to explain in detail，how can we eliminate -pi^2 directly on the right and the left side have no change

[minwithoutintroduction]

رائع جدا.
أول فيديو في القناة اشاهده.أعجبني كثيرا.
متتبعك من مدينة تنغير بالمملكة المغربية

[benphillips]

How can you set (x + pi)(x - pi)... = 0, then set it equal to sin(x)/x?

[diellzagerguri]

Thank you for this proof, really useful. May i ask where you got it from - which book?

[Packerfan]

how do you go from

sinx / x = (x^2 - pi^2)(x^2 - 4pi^2)...

to

sinx / x = (1 - x^2/pi^2)(1 - x^2/4pi^2)...

where did the -pi^2 in each factor here go?

[trumplostlol]

So, someone has proved that the series will converge after expansion? Or do we need to assume x tends to zero?

[מאמין-גנ]

Thank you so much !! that was very clear !!

[sesppsfd]

why do we need to make the factors into the form of (1-x^2/(npi)^2) but not keeping (x^2-(npi)^2)?

[lbqxylk]

Perhaps another commenter noted this already but you should have

sin(x)/x = (1-x/π)(1+x/π)(1-x/2π)...

Now when you take the limit as x approaches 0, you get 1 on both sides. As you had it, you would get 1 on the left side and an undefined infinite product on the right hand side. In particular, on the right side you'll have the infinite product

π(-π)(2π)(-2π)...

after you take the limit as x tends to 0.

[EtotheiPi]

The town is called Buhsel, not Beisel.

[josepcolominas]

Factorization theoreme implies a not justified multiplicative constant ?

[yuvraj]

It's not prononuciated Zahlen but "Sahlen"

[adwz]

to prove carefully, u need to even prove the intermediary theorems, such as the Weierstrass product formula; this wasn’t done