Consider the fourier series
$$\cos(\theta)+\frac{\cos (3 \theta)}{9}+\frac{\cos(5 \theta)}{25}+\cdots.$$
Find the function which has this Fourier representation.
Answer: The fourier series can be written as $\sum_{n=0}^{\infty} \frac{\cos(2n+1)\theta}{(2n+1)^2}$.
I know that $ f(x)=x , \ 0 \leq x \leq \pi $ has fourier series $ f(x) \sim \frac{\pi}{2}-\frac{4}{\pi} \sum_{i=2n+1}^{\infty} \frac{\cos(2n+1) \theta}{(2n+1)^2} $.
So the function $ f(x)=\frac{\pi}{4}(\frac{\pi}{2}-x) $ has the fourier series $\sum_{n=0}^{\infty} \frac{\cos(2n+1)\theta}{(2n+1)^2} $.
Hence the function should be $ f(x)=\frac{\pi}{4}(\frac{\pi}{2}-x) $.
I need confirmation of my work. Any better method ? Is there any help ?