I gave a presentation for a seminar class yesterday on Fourier analysis, and introduced the sawtooth function as a counterexample, for a function whose Fourier series is not termwise differentiable. This example was motivated by my claim that physicists run around willy-nilly, and differentiate term-by-term without regard for convergence.
So, in short, if $f(x)=x$ for $0\le x\le 2\pi$ and we extend by periodicity, then $f$ has Fourier series $\sum_{n\neq 0, n\in\mathbb{Z}}^\infty -\frac{1}{in} e^{inx}$, which converges pointwise (conditionally). If we differentiate termwise, we get the series $-\sum_{n\neq 0} e^{inx}$, which never converges, and yet we should have gotten $1$ off of $2\pi\mathbb{Z}$.
My point: just after my presentation, a friend with a greater inclination to physics actually showed a "physicist's justification" of the above series actually equaling $1$. I.e. if we pretend that the geometric sum $\sum_{n=0}^\infty r^n$ actually converges for $|r|=1, r\neq 1$ then we can write the above series as $-\sum_{n=1}^\infty e^{-inx}+e^{inx}=-\frac{e^{-ix}}{1-e^{-ix}}-\frac{e^{ix}}{1-e^{ix}}$, which actually equals $1$ for $x\notin 2\pi\mathbb{Z}$!
My question is: is there a formal justification for this manipulation? I have in mind one involving the power series ring $\mathbb{C}[[x]]$, so that we can formally ignore convergence issues. The fact that this method, which is so simple, actually gives the right answer has convinced me that there must be some truth to it. Thanks for any advice or references you may be able to provide!
This is a valid manipulation of formal Laurent series $\mathbb{C}[[e^{ix},e^{-ix}]]$. Note that formal Laurent series don't form a ring because some pairs of elements can't be multiplied, but we indeed have that $\sum_{k=0}^{\infty}{e^{ikx}}$ is the multiplicative inverse of $1-e^{ix}$ and similarly for negative powers.