Fourier series for $f(x)=x^{2m}$ on $[-\pi; \pi]$

Question

Am I right in the Fourier series on $[-\pi; \pi]$? $$x^{2m}\sim\frac{\pi^{2m}}{2m+1}+2\sum\limits_{n=1}^{+\infty}\sum_{k=0}^{m-1}\frac{(-1)^k\pi^{2(m-k-1)}}{n^{2(k+1)}(2m-2k-1)!}\cos\pi n\cos nx$$