# Fourier Series Convergence

Going over some revision. Not really sure what to do for the last bit of aii)

I know at $x = 0$, it will converge to $0$ and at $x = \frac{M}{2}$ it will converge to $1$, I'm not seeing how this relates to answering the last bit. Thanks!

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How do you know it converges to $\frac{3}{2}$ at $x=0$? I think it shouldn't. – Matt L. Apr 22 '13 at 15:29
Sorry, I meant it converges to zero, I was looking at a different question and getting muddled up! – Mike Miller Apr 22 '13 at 15:36
OK, at discontinuities the Fourier series converges to the mean of the values to the left and to the right of the discontinuity, i.e. at $x=0$ it indeed converges to 0. – Matt L. Apr 22 '13 at 16:06

## 1 Answer

It should converge everywhere $f$ is continuous to the value of $f$ here. This gives the result you need. As far as I can see, they're just emphasizing that it works in the middle but not at the endpoints. The "hence" seems misleading.

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I can't see how just plugging $M=20$ in shows that works though. I feel I'm not understanding something fundamental here. – Mike Miller Apr 22 '13 at 15:57
The choice of $M$ has nothing to do with anything; the only variable which matters is $x/M$. I think the question is very poorly phrased. The only thing there is to prove is that the Fourier series converges, which you can't deduce from its value at two points, as far as I can see – Sharkos Apr 22 '13 at 16:07
So what would I do then? Quite confused now as to what I have to show. – Mike Miller Apr 22 '13 at 16:14
Well, as I say, the question is very poorly worded. If you can assume the result that piecewise continuous functions like this have convergent Fourier series away from discontinuities (where you get the average of the two sides), then the result is trivial. Otherwise, it has nothing to do with the rest of the question, and it's bizarre. I guess I can't help with the interpretation. – Sharkos Apr 22 '13 at 16:36
Well thanks anyway! – Mike Miller Apr 22 '13 at 17:40