Differentiation of Fourier Series

Question

The Series: $$\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos(2n-1)x}{(2n-1)^2}$$ is the Fourier cosine series for the function $f(x)=x$ on the interval $0<x<\pi$. Differentiate this series term by term to obtain a representation for the derivative $f^{'}(x)=1$ on that interval. State why the procedure is reliable here.

Please help. I do not understand this question. Thanks

Answer 1

There are two parts of the question:

1. Differentiate (i.e. take the derivative of) the series by differentiating the expression to the right of the sum symbol.
2. Explain why the derivative of the sum (with an infinite number of terms) equals the sum of the derivatives. This is not always valid so find a suitable theorem in your course book.