Convergence properties of series $\sum (a_n\cos nx+b_n\sin nx)$ 
I did Part I using Cauchy criterion for convergence.
I have no idea how to do Part II and III though. 

(II): First do it when $k=1$. There is a natural candidate for the derivative Then give an estimate of the difference between $\frac{f(x+h)-f(x)}h$ and $\sum_k k(b_k\sin(kx)+a_k\cos(kx))$, where $f$ is the sum of the Fourier series. The diea is similar for $k>1$.
(III) Justify the differentiation under the series.
