Hello,
This is a question using Fourier series to show that $t^2$ = $f(t)$ in Fourier series.
I could do the Fourier series part but I am not sure how to do the deduce = $\pi^2/3$ part of the question?
Could somebody help me out?
2026-05-15 11:50:06.1778845806
How to do Fourier series, deduce question?
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Hint:
Can you find two expressions for $f(0)$?
We have $f(t)=t^2$ and $f(t)=\frac{\pi^2}{3}+4\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^2}\cos(nt)$ and they give the same value for $t=0$. Then after equating these two expression, re-arrange and deduce that $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^2}=\frac{\pi^2}{12}.$