# Math Help - Fourier Series problem

1. ## Fourier Series problem

find a half-range series of $f(x)=x^{2} \ \ in \ \ (0,1)$

and then show that:

$\sum_{n=1}^{\infty}{\frac{1}{n^{2}}}=\frac{\pi^{2} }{6}$

i got a series:

$x^{2}=\frac{4}{3}+\sum_{n=1}^{\infty}{\frac{4(-1)^{n}}{n^{2}\pi^{2}}\cos(n\pi x) }$

2. Originally Posted by silversand
find a half-range series of $f(x)=x^{2} \ \ in \ \ (0,1)$

and then show that:

$\sum_{n=1}^{\infty}{\frac{1}{n^{2}}}=\frac{\pi^{2} }{6}$

i got a series:

$x^{2}=\frac{4}{3}+\sum_{n=1}^{\infty}{\frac{4(-1)^{n}}{n^{2}\pi^{2}}\cos(n\pi x) }$
How did you get the lead term $\frac{4}{3}$ ?

3. Originally Posted by silversand
find a half-range series of $f(x)=x^{2} \ \ in \ \ (0,1)$

and then show that:

$\sum_{n=1}^{\infty}{\frac{1}{n^{2}}}=\frac{\pi^{2} }{6}$

i got a series:

$x^{2}=\frac{4}{3}+\sum_{n=1}^{\infty}{\frac{4(-1)^{n}}{n^{2}\pi^{2}}\cos(n\pi x) }$