I am trying to obtain a result in condensed matter physics, and got stuck in math. Basically I want to Taylor expand the coefficients of a Fourier series expansion of a function $f$: $$f(x)=\sum_n c_n e^{inx}$$ with $$c_n=\frac{1}{2\pi}\int_{-\infty}^{\infty} f(x) e^{-inx} dx.$$ Then I try to substitute $f(x)=\sum_k \frac{1}{k!}\frac{d^k}{dx^k}f(0) x^k$ in the integral of $c_n$. I get $$c_n=\sum_k \frac{1}{2\pi}\frac{1}{k!}\frac{d^k}{dx^k}f(0)\int_{-\infty}^{\infty} x^k e^{-inx} dx.$$
Now, if I could integrate that, I would be able to advance. Have I done something wrong in the derivation so far? If I have not, how can I obtain the result of that integral?
Edit: A more complete statement of the problem follows. I have the operator A(p,r) which is periodic: $$A(p,r+T)=A(p,r)$$ And can be written as: $$A(p,r)=\sum_G A_G(p)e^{iG.r}$$
I also have a function:
$$\Psi_{n,k} (r) = e^{i k.r} u_{n,k}(r)$$
Where $u_{n,k}$ is periodic with same period T and can be expanded in a Fourier series: $$u_{n,k}(r)=\sum_G Y_{n,k,G} e^{i G.r}$$
According to the textbook I can prove, by Taylor expanding the coefficients $A_G(p)$ that the action of the operator A acting on $\Psi$ can be written as: $$A(p,r)\Psi_{n,k}(r)=e^{i k.r}U_{n,k}(r)$$
Where $U_{n,k}$ is a linear combination of the $u_{n,k}$.
(Please let me know if this is not the correct way to format this)