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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

Yes, the general form for a simple cf is

**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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**zetafunc.****Guest**

But how do you show that

and

are the same?

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

They are not the same obviously but we might be able to prove they both converge to the same thing.

**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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**zetafunc.****Guest**

They said something about evaluating that first fraction at every second term, to get the second fraction...

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

And how does that prove they are the same?

**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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**zetafunc.****Guest**

It doesn't but they are claiming there is a way to get the simple CF from the first one...

Or maybe just someone on Wikipedia typed it for the hell of it? Maybe that's why they didn't elaborate on what they meant exactly...

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

I do not see any way right now of deriving one from the other.

But I can prove they both converge to the same thing.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**zetafunc.****Guest**

What methods of proof do you use?

Also, still nothing from adriana...

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

Consider her dead meat.

It is possible to prove what they converge to by algebra and a little trickery.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**zetafunc.****Guest**

You may be right... this could venture into F territory. Something is wrong. Just a pity it ended so abruptly.

You mean trying to generate the CF?

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

No;

Take a look at the first cf and call it x

*Last edited by bobbym (2013-02-25 07:27:21)*

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**zetafunc.****Guest**

Sorry, I had to take my sister to my grandmother's house.

I see, you just get the fraction on its own, invert it, and repeat, and you end up with a quadratic which has a root at x = √3. I imagine the same thing would work for the other CF.

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

Hi;

when you solve for x you get √3

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**zetafunc.****Guest**

Yes, that is what I meant -- eventually when you follow those steps, you get the original fraction (x again).

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

Yes, somewhere in th cf there will be a repeat of x. You just replace it and solve for x.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**zetafunc.****Guest**

That seems pretty handy to check if these Wikipedia CFs converge as they're supposed to... but, I suppose this method might be hairy if you have a long period, for large n, say.

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

It is easier to use numerical methods and arrive at an experimental result.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**zetafunc.****Guest**

We looked at continued fractions and applying them to solve Pell's equation, are there any applications of infinite nested square roots? Those seem interesting too.

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

I do not know of any offhand apps but they are solved in the same way as a cf.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**zetafunc.****Guest**

Yes, I have read about them. The Wiki article is a lot shorter than the CFs one though.

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

They are less useful apparently. CF's are very big in number theory and numerical analysis.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**zetafunc.****Guest**

What other things are they used for in number theory?

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

Just for the Fermat - Pell equation as far as I know. But these are an important class of diophantine equations. They are more important I think in numerical analysis.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**zetafunc.****Guest**

A girl asked me what I was doing today in chemistry, I was trying to solve a Pell equation using one of my CFs. When I tried to show her what a CF was, she said "I'd rather paint my nails".

23:53, nothing from adriana at all. Sigh...

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,376

Yikes, an intellectual!

I got another method for CF's.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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