Manipulating the Fourier series of $x^2$ to find the sum of a particular series

Question

The first part of the question was to prove the Fourier series is $$\sum\frac{4(-1)^n}{n^2} \, \cos(nx) $$ on interval $[-\pi,\pi]$ which I had no problem with. The next question was to find the sum of the series $$\sum\frac{-1^{n+1}}{n^2}. $$ It says to set $x=0$, which is kinda obvious but how would I go about finding the sum of that series by from the Fourier series. I read online about Parsevral's theorem which relates a sum of the Fourier coefficients squared to an integral. But from what I've seen I cannot use this from my case because it just gives me info about what the series $$\sum\frac{{16}}{n^2} $$ converges to. Please tell me if I am wrong. the answer is $\pi^2/12$ from the back of the text book. Sorry for obvious lack of usage with the MathJax I just really needed help.

Answer 1

• Find the correct Fourier series.
• Check whether the series represents the given function in the interval $[{-\pi},\pi]$, according to the theorems in your book.
• Draw conclusions.

Answer 2

Put $x=0$ in $f(x)=\sum \frac{4(-1)^{n}}{n^2}\cos x$ or whatever the Fourier sum is. You get $$f(0)=-4\sum \frac{(-1)^{n+1}}{n^2},$$ from where you obtain the sum.