What is the proof for the value of the Riemann Zeta Function at 2?

[imranq]

So we were going over geometric series in my calc class (basic, I know), however I was intrigued by one point that the prof. made during lecture

$$\frac{\pi^2}{6} = \sum^{\infty}_{n=1}\frac{1}{n^2} = \zeta (2)$$

That's amazing (at least to me). Looking for the explanation for this, I found a bunch of stuff relating to Fourier analysis which was - unfortunately - written in vague terms. Would someone explain this proof that is accessible to a Calc II student? Thanks

 

[Hurkyl]

Wikipedia offers an elementary proof:

http://en.wikipedia.org/wiki/Basel_problem#A_rigorous_proof​

but I highly doubt there is an 'easy' elementary proof of this statement.

 

[imranq]

That was fast and I understand it now, thanks! By the way, what is the Fourier series used for?

 

[mathwonk]

you might enjoy the computation in euler's precalculus book of this series. he also did it for many other values of zeta, and apparently gave a general formula relating all even values of zeta to the bernoulli numbers, as given in the appendix to milnor and stasheff, characteristic classes, using Fourier series.

 

[yasiru89]

http://www.maths.ex.ac.uk/~rjc/etc/zeta2.pdf
Gives 14 different evaluations (its also linked on the wikipedia page incidentally).

The part on zeta function values at http://www.ams.org/bull/2007-44-04/S0273-0979-07-01175-5/S0273-0979-07-01175-5.pdf, guides you through Euler's original derivation (which is easily made rigorous once the infinite product for the sine is obtained).

Fourier series methods offer shorter and easily generalised proofs, for instance, check the following threads,

https://www.physicsforums.com/showthread.php?t=192765
and
https://www.physicsforums.com/showthread.php?t=95994

There was also an elementary paper by Xuming Chen in the College Mathematics Journal called 'Recursive Formulas for [itex]\zeta(2k)[/tex] and [itex]L(2k - 1)[/tex]'.<br /> <br /> Hope this helped![/itex][/itex]

 

[imranq]

http://www.maths.ex.ac.uk/~rjc/etc/zeta2.pdf
Gives 14 different evaluations (its also linked on the wikipedia page incidentally).

The part on zeta function values at http://www.ams.org/bull/2007-44-04/S0273-0979-07-01175-5/S0273-0979-07-01175-5.pdf, guides you through Euler's original derivation (which is easily made rigorous once the infinite product for the sine is obtained).

Fourier series methods offer shorter and easily generalised proofs, for instance, check the following threads,

https://www.physicsforums.com/showthread.php?t=192765
and
https://www.physicsforums.com/showthread.php?t=95994

There was also an elementary paper by Xuming Chen in the College Mathematics Journal called 'Recursive Formulas for [itex]\zeta(2k)[/tex] and [itex]L(2k - 1)[/tex]'.<br /> <br /> Hope this helped![/itex][/itex]

[itex]$<br /> Thanks a lot!$[/itex]