Sum to infinity of recipricol of squares

Can anyone show me how to get the sum to infinty of: 1/1² + 1/2² + 1/3²...

My friend says its ?²/6, which looks about right but I have no idea how to get there.

Thanks

It's an extremely interesting result.
http://en.wikipedia.org/wiki/Basel_problem

The taylor series can be used. Very briefly it states:
$f(x)=f(0)+xf'(0)+\frac{x^{2}}{2!}f''(0)+\frac{x^{3}}{3!}f'''(0)+....f^{r}(0)\frac{x^{r}}{r!}+....[br]$

That's Euler's argument and wouldn't really count as a proof these days - you'd need to be more rigorous regarding convergence etc.

I'm reading some introductory analysis [eg $|a_{n}-l|<\epsilon \textit{ for every }n>N \textit{ implies } a_{n}>l\textit{ as }n>\infty]$ and can only imagine how hard it must be to produce a proper rigorous proof.

Well some of the claims in the proof are wrong. It is true that sinx/x equals the given infinite product but one cannot claim generally that a function's power series equals an infinite product of factors containing the function's roots - for example look at the exponential series

e^x = 1 + x + x^2/2! + ...

yet the function e^x has no roots, not even counting complex ones. Yet by Euler's reckoning these "roots" should add to -1.

Well some of the claims in the proof are wrong. It is true that sinx/x equals the given infinite product but one cannot claim generally that a function's power series equals an infinite product of factors containing the function's roots - for example look at the exponential series

e^x = 1 + x + x^2/2! + ...

yet the function e^x has no roots, not even counting complex ones. Yet by Euler's reckoning these "roots" should add to -1.

You can't actually critise Euler's reasoning and belief, as they were the earlier years of mathematics, rather admire and correct him.

It seems that the broadening and discovering of more math has come to a halt.

Interesting. Using infinite series to prove properties seems to require delicate work. Is there a way to prove $e^{i\theta}=\cos \theta + i\sin \theta$ other than using power series?

More often than not sine, cosine and e^z are defined as power series to start with and the result follows from the definition.

Also you can define sine and cosine as complex exponential functions anyway, and the result is again obvious. It is all dependant on the definition.

1/1² + 1/2² + 1/3²...

=&#950;(2)=(&#960;²/6)

Newton.

But it wouldn't do them any harm to learn!

Anyone who can find the value of &#950;(3) will get their own little piece of fame

It's known to be irrational and that is all I think. A result due to Apery and so zeta(3) is often called Apery's constant.

I guess most students see a proof of zeta(2) = pi^2/6 through Fourier Series.