Complex Fourier series of $f(\theta) = e^{\theta}$
I have the following Fourier series problem:
> Let $f(\theta)$ be the periodic function such that $f(\theta) = e^\theta$ for $-\pi<\theta\leq\pi\;$, and let $\;\displaystyle\sum_{-\infty}^{\infty}c_ne^{in\theta}$ be its Fourier series; thus $e^\theta = \displaystyle\sum_{-\infty}^{\infty}c_ne^{in\theta}$ for $\;|\theta|<\pi$. If we formally differentiate this equation, we obtain $e^\theta = \displaystyle\sum_{-\infty}^{\infty}inc_ne^{in\theta}$. But then $c_n = inc_n$, or $(1-in)c_n = 0$, so $c_n=0$ for all $n$. This is obviously wrong; where is the mistake?
Any ideas / hints?...
The snag is that the derivative of the Fourrier series of $e^\theta$ is not convergent.