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What does one have to prove/show in order to justify $ f(x)=\sum_{n=1}^k c_n \sin(nx) $ has derivative f'(x)=\sum_{n=1}^k c_n n \cos(nx) ?

I am used to assuming this to be true. Say also that $\sum\limits_1^\infty |c_n| < \infty$. Thank you.

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    In the case $k = \infty,$ a sufficient condition (and one that you probabaly already know) is that the sums $\sum c_n \mathrm{sin}(nx), \sum c_nn\mathrm{cos}(nx)$ converge uniformly on compact sets.2011-09-19

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