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.
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.