I'm tutoring a set of problem sheet to do with Fourier series and one problem is as follows:
The Fourier series for a sawtooth wave is,
$f(x)=x=-\sum^{\infty}_{n=1}\frac{2(-1)^n\sin(nx)}{n}$ for $-\pi < x<\pi$.
If you differentiate this you get
$1=-2\sum^{\infty}_{n=1}(-1)^n\cos(nx)$ again for $-\pi < x<\pi$
What is wrong with this?
I have the solutions sheet and it says that it does not converge to 1 (fair enough, I plotted it to large $n$ and it sort of converges but oscillates between 0 and 2 in the interval) and then states ...
An assumption has been made that you can interchange the order of summation and differentiation in the result stated.
It then goes to note that you can interchange the order of summation and integration
I don't understand the argument, can anyone shed some light on this?