Telemachus
- 820
- 30
Hi. Well, I have to demonstrate that \sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}=\frac{\pi^2}{8}
Using a previous result of a Fourier series for f(t)=1 if 0<t<1, f(t)=0 if -1<t<0.
I've found that:
f(t)\sim{}1/2+\sum_{n=1}^{\infty}\frac{2}{(2n-1)\pi}\sin([2k-1]\pi t)
I think this result, the series expansion for the function is right. But when I've tried to demonstrate what the problem asks me I get to an absurd, so I'm not sure where the error is.
This is what I did:
S(1/2)=1=\frac{1}{2}+\sum_{n=1}^{\infty}\frac{2}{(2n-1)\pi}\sin([2k-1]\frac{\pi}{2})
Working from here:
\frac{1}{2}=\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{(2n-1)}
\frac{\pi}{4}=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{(2n-1)}
\frac{\pi^2}{16}=\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}
So, I'm doing something wrong. Perhaps you can see more clearly where am I committing the mistake.
Using a previous result of a Fourier series for f(t)=1 if 0<t<1, f(t)=0 if -1<t<0.
I've found that:
f(t)\sim{}1/2+\sum_{n=1}^{\infty}\frac{2}{(2n-1)\pi}\sin([2k-1]\pi t)
I think this result, the series expansion for the function is right. But when I've tried to demonstrate what the problem asks me I get to an absurd, so I'm not sure where the error is.
This is what I did:
S(1/2)=1=\frac{1}{2}+\sum_{n=1}^{\infty}\frac{2}{(2n-1)\pi}\sin([2k-1]\frac{\pi}{2})
Working from here:
\frac{1}{2}=\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{(2n-1)}
\frac{\pi}{4}=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{(2n-1)}
\frac{\pi^2}{16}=\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}
So, I'm doing something wrong. Perhaps you can see more clearly where am I committing the mistake.