Let $\{c_n\}$ be a sequence of complex numbers converging to zero.
Let $r \in \mathbb{R}$ with $ 0<r<1$.
We can show that $$f_n = \sum_{n = -N}^{N}c_n r^{|n|}e^{inx} $$ converges uniformly over $[-\pi, \pi]$.
This can be done by noting that $\{c_n\}$ is convergent and therefore bounded uniformly. And that $\sum_{n = -N}^{N}r^{|n|}e^{inx}$ converges uniformly by the Weierstrass M-test, so $\sum_{n = -N}^{N}c_n r^{|n|}e^{inx} $ converges uniformly.
How can we show that the sequence of derivatives $$f'_n =\sum_{n = -N}^{N}inc_n r^{|n|}e^{inx} $$ converges uniformly over $[-\pi, \pi]$?
Is it uniformly convergent?