A question on convergence of derivative of power series

Question

This is a question from Fourier Analysis with Applications by Folland. First we write Fourier series for $$e^{\theta}=\sum c_ne^{in\theta}$$ We differentiate this series term by term to obtain $$e^{\theta}=\sum inc_ne^{in\theta}$$ But by uniqueness of Fourier series we have $c_n=ic_n$ which implies that $c_n=0$. This is clearly wrong but where is the fallacy. If someone can tell what is going wrong it would be great. Thanks.