Exercise 1.11 Let \( f \in C^{\infty}(\mathbb{R} / \mathbb{Z}) \) and let \( c_{k} \) be its Fourier coefficients, Show that the sequence \( c_{k} \) is rapidly decreasing; i.e., for each \( N \in \mathbb{N} \) there is \( d_{N}>0 \) such that for \( k \neq 0 \),
\[
\left|c_{k}\right| \leq \frac{d_{N}}{|k|^{N}} .
\]
(Hint: Compute the Fourier coefficients of the derivatives of \( f \).)