prove that series

**dhiab** · November 29th 2009, 10:43 PM

prove that :
$\sum_{n=1}^{\infty }(\frac{1}{n^{2}})=\frac{\pi ^{2}}{6}$

**simplependulum** · November 29th 2009, 11:40 PM

Originally Posted by dhiab

prove that :
$\sum_{n=1}^{\infty }(\frac{1}{n^{2}})=\frac{\pi ^{2}}{6}$

Consider $\sin(x) = x \prod_{n=1}^{\infty} \left ( 1 - \frac{x^2}{(n\pi)^2} \right )$

Then compare the coefficient of $x^2$ with its Taylor's series .

**chisigma** · November 30th 2009, 01:03 AM

Originally Posted by simplependulum

Consider $\sin(x) = x \prod_{n=1}^{\infty} \left ( 1 - \frac{x^2}{(n\pi)^2} \right )$

Then compare the coefficient of $x^2$ with its Taylor's series .

... $x^2$ or $x^3$ ?

...

Kind regards

$\chi$ $\sigma$

**simplependulum** · November 30th 2009, 01:13 AM

Oh , yes

If i say comparing with the series , it should be $x^3$ , but not $x^2$ .

Or say compare the coefficient of $x^2$ after dividing the factor $x$

**tonio** · November 30th 2009, 01:54 AM

Originally Posted by dhiab

prove that :
$\sum_{n=1}^{\infty }(\frac{1}{n^{2}})=\frac{\pi ^{2}}{6}$

Take a peek at this very nice paper: http://www.math.titech.ac.jp/~inoue/...lder/zeta2.pdf
You have there no less than 14 different proofs of the result $\zeta(2)=\sum\limits_{n+1}^{\infty}\frac{1}{n^2}=\ frac{\pi^2}{6}$ , from rather basic ones to those that require more knowledge....enjoy!

Tonio

**Krizalid** · November 30th 2009, 01:56 AM

http://www.secamlocal.ex.ac.uk/peopl.../etc/zeta2.pdf

**Jameson** · November 30th 2009, 01:59 AM

I believe the OP mostly posts challenge problems but places them in the regular forums. This thread has been answered many times over and I am going to PM the OP about using the Challenge Problems forum to save everyone time.

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