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:

$x^{2}=\frac{4}{3}+\sum_{n=1}^{\infty}{\frac{4(-1)^{n}}{n^{2}\pi^{2}}\cos(n\pi x) }$
you should get 1/3 (a0 is 2/3 and you want half of it)

see the following page for explanation (take l = 1 in your problem)
half range fourier series of f(x) = x in (0,l)

for the deduction
put x = pi

for the formulae use http://keral2008.blogspot.com/2009/0...er-series.html