I am studying Fourier Analysis from the M.Stein book, in Chapter 3 there is a whole few pages dedicated to showing that the Fourier series of the sawtooth function defined as $$f(x)=i(\pi -x)$$ odd in x and defined between $0<x<\pi$ diverges at $x=0$.

The Fourier series of this function is $$\sum_{n\neq 0}\frac{e^{inx}}{n}$$.

I'm not quite understanding in what sense the series diverges. The symmetric partial sums of the series are all equal to $0$, so the series should be conditionally convergent to $0$ at $x=0$. I don't understand in what sense this series diverges.



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