How can I expand $\sin(x) + x$ on Fourier series in $[0, 2\pi]$ for $2\pi$ period?
This what I got from Euler-Fourier formula:
$a_0 = 2\pi$
$a_n = 0$
$b_n = \frac{-2}{n}$
This is final result that I've calculated: $$\pi - 2 \sum_{n=1}^{\infty}\frac{1}{n}\sin{nx}$$
Unfortunately I see that it doesn't approximate $x + \sin{x}$ correctly.