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#8226 2013-02-21 02:20:00

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,302

Re: Linear Interpolation FP1 Formula

Hi;

She is having a good time. You do not have to think about it or her.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8227 2013-02-21 02:54:05

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

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?

#8228 2013-02-21 03:01:31

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,302

Re: Linear Interpolation FP1 Formula

She will be back soon, so relax and do some math.

Your series is known as the Grandi Series. Look here first:

http://en.wikipedia.org/wiki/Grandi%27s_series


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8229 2013-02-21 03:05:55

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

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...

#8230 2013-02-21 03:08:35

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,302

Re: Linear Interpolation FP1 Formula

Remember the Grandi does not converge. It is convergent in a Borel sense. Which loosely means we would like it to converge to 1 / 2, here are the usual methods.

http://en.wikipedia.org/wiki/Summation_ … 27s_series

Please your post your method because I am not seeing it there.

Last edited by bobbym (2013-02-21 03:08:59)


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8231 2013-02-21 03:12:27

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Then let x = 0 for example.

#8232 2013-02-21 03:25:37

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,302

Re: Linear Interpolation FP1 Formula

Hi;

The second series does converge to 1 / 2.

Last edited by bobbym (2013-02-21 03:26:31)


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8233 2013-02-21 03:27:58

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

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.

#8234 2013-02-21 03:28:34

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Sorry, just saw your edit...

So it does converge? Why not at x = 0?

#8235 2013-02-21 03:33:34

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,302

Re: Linear Interpolation FP1 Formula

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 (2013-02-21 04:06:05)


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8236 2013-02-21 04:47:34

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

So how do we know for certain whether or not the Fourier series for a particular function converges or not? I did read somewhere that it 'almost completely' converges in the interval...

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

Also, I think I will need to go for UCL's accommodation, they said I have to wait at least 4 years for a flat...

#8237 2013-02-21 04:55:05

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,302

Re: Linear Interpolation FP1 Formula

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.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8238 2013-02-21 05:06:37

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Either halls of residences or just flats provided by UCL, on their campus.

Can you give me a demonstration of Gibb's for a simple Fourier series?

#8239 2013-02-21 05:07:05

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

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

#8240 2013-02-21 05:08:06

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,302

Re: Linear Interpolation FP1 Formula

It is hard. I do not get it either. Worse, I have never found a book that was willing to explain it.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8241 2013-02-21 05:12:49

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

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...

#8242 2013-02-21 05:19:39

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,302

Re: Linear Interpolation FP1 Formula

From a practical standpoint you can use the idea of probably convergent except at jump discontinuities.

I have to go pay a bill I will be gone for several hours. See you then.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8243 2013-02-21 05:24:26

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

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

Okay, see you later.

#8244 2013-02-21 10:20:48

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,302

Re: Linear Interpolation FP1 Formula

Not necessarily, Sometimes you can use a truncated part of a divergent series for real numerical work.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Offline

#8245 2013-02-21 10:37:07

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

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.

#8246 2013-02-21 10:41:45

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,302

Re: Linear Interpolation FP1 Formula

Facebook got hacked yesterday or this morning.

I will look for an example.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Offline

#8247 2013-02-21 10:42:53

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

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.

#8248 2013-02-21 11:10:10

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Found a photo of her and her BF cuddling, from 3 days ago. I guess we know what decision she made... I would not be surprised if she decides not to contact me again. Oh well.

#8249 2013-02-21 12:14:31

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,302

Re: Linear Interpolation FP1 Formula

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.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Offline

#8250 2013-02-21 12:26:01

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Yep. I suppose it just reinforces the fact that I shouldn't get involved with her and just keep her as a friend or acquaintance. My time will hopefully come soon, just with another woman. She might not exist but I won't lose hope, hope makes the journey worthwhile.

Thank you, that would be really helpful.

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