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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?

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    This may help: http://www.math24.net/differentiation-and-integration-of-fourier-series.html2012-01-27
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    Also, see page 27 onward here: http://www.google.com/url?sa=t&rct=j&q=integration+fourier+serie&source=web&cd=12&ved=0CDEQFjABOAo&url=http%3A%2F%2Fwww.math.umn.edu%2F~olver%2Fam_%2Ffs.pdf&ei=Ch4jT-ryGMmftwe6oZCiCw&usg=AFQjCNF9TCrgqGfIVgDxlnUuknISwbiL8Q2012-01-27
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    If you differentiate once more, you will obtain a linear combination of linearly independent functions equal everywhere to zero. It contradicts to the linear independence of functions.2012-01-28
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    Looking at the links David left, pg. 670 of the Pete Olver textbook chapter pdf has a good answer. It seems that it is because, even though the limits are stated, the series is still converging to the sawtooth and not x, so the differentiation is converging to the differential of the sawtooth, i.e. $1-2\pi\delta(x-\pi)$. I had considered this but assumed that because in the question was explicit about the limits, this was not an issue and would converge to x ...2012-01-28

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