I sort of miss her in a friend sort of way, not a romantic sense. You know, just having someone to talk to...

Also, I have arrived at 1 - 1 + 1 - 1 + ... = 1/2 using Fourier series (I'm aware that the series diverges so this is not really correct), but I don't know if what I did was gibberish, could you check it?

Yes, I have read that -- but basically I just found the Fourier series of x/2, differentiated both sides and let x = 0, pi/6, and other values which resulted in 1/2 = 1 - 1 + 1 - 1 ... etc. but I don't know if that type of manipulation is valid. If I'm getting Grandi's series on the RHS, then that means the Fourier series doesn't converge at those points...?

I need to go in a few minutes to apply for my flat...

Okay, so we cannot just differentiate a FS and expect everything to work out... so, if my FS is 2pi-periodic (say, over the integral [-pi,pi], then does it converge for all values of x, -pi < x < pi? (My original FS for x/2, not the differentiated one)

I have to go now, I will be back in an hour or so.

Hi;

Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue...

At 0 it is the Grandi Series.

The Fourier series does not always converge, and even when it does converge for a specific value x0 of x, the sum of the series at x0 may differ from the value f(x0) of the function. It is one of the main questions in harmonic analysis to decide when Fourier series converge, and when the sum is equal to the original function.

Fourier series do not always converge.

In numerical work we presume that the series would converge in the interval except at discontinuities. There is something called Gibb's phenomena which describes this.

Last edited by bobbym (2013-02-22 03:06:05)

You mean dorms?

I did read somewhere that it 'almost completely' converges in the interval...

Except at discontinuities which is quite often because those are the type it is often used on.

So you're saying Gibb's phenomena would tell me whether or not a Fourier series converges

No, Gibb's shows it will not always converge in the interval. To really know would require in a practical sense numerical experimentation. Else, you would need Harmonic analysis. I choose the first one.

Harmonic analysis looks interesting but difficult too, I doubt I will understand much of it...

I think it is offered as either a third year course or a graduate-level course. I don't know what the prerequisites are though...

But even if I get a pretty-looking series, it means nothing mathematically, because it requires proof of the convergence...

Okay, see you later.

Do you have an example?

Also, I just found adriana on Facebook (I don't have an account on there though, I was just curious). Apparently she updated it yesterday... I did not hear from her though.

Really? It says she updated 23 hours ago (21:40-ish on the 21st of Feb)... which would coincide with when she returned to London.

Okay, thank you.

Cuddling? I thought she was the other way. It is a real shame that the story of Pinocchio is not true. Could you imaging how many big noses there would be?

Have an example and will latex it tomorrow.