Suppose we have the Fourier Series :
f(x)=$\sum_{k=1}^{\infty} C_k f_k(x)$=$\sum_{k=1}^{\infty} C_k \sin(kx)$ defined in $(a,b) \in \mathbb{R}$
Using Dirichlet criterion I have shown the sum is continuous in $ [ a, b ]$. Now I need to show that the sum $ f\in \mathcal{C}^i (a,b) $.
Is it sufficient to apply Dirichlet criterion (i times) for the sum of the partial derivatives of every $f_k$?