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**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 291

Hi;

[latex fixed by admin]

Fn is the nth Fibonacci number

*Last edited by ElainaVW (2013-05-13 19:34:11)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

Hi;

Please use the latex capabilities of M or the codecogs site for perfect latexing every time.

**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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,507

Hi ElainaVW

I am getting

.Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 291

Hi anonimnystefy;

Thanks for the kind solutions.

*Last edited by ElainaVW (2013-05-14 22:21:46)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

Hi;

Yes, the integration is too tough. But that is not the only way to skin a cat.

**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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**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 291

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

Hi ElainaVW;

Congratulations on your new member status.

**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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**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 291

Hi Bobby;

Thanks, I had two of the best there are over here.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

Hmmmm.

Please use my username.

Usually, accelerators like sequences where e is shrinking by a fixed constant.

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

Combinatorics is Algebra and Algebra is Combinatorics.

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Hi bobbym, How did you do that?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

Hi Agnishom;

Bother the OP? It was easy, I am good at being annoying.

But besides from that which do you mean?

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

Combinatorics is Algebra and Algebra is Combinatorics.

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The Summation

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

Hi Agnishom;

There are many ways to do that sum. When you have the gf you have the sum of the terms of the sequence the gf represents. Normally, the gf is treated formally but when you use it for summation the rules of power series apply,

The numerical way just takes advantage of the double rule and the principles outlined over here.

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

Combinatorics is Algebra and Algebra is Combinatorics.

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bobbym wrote:

When you have the gf you have the sum of the terms of the sequence the gf represents. Normally, the gf is treated formally but when you use it for summation the rules of power series apply,

I know nothing about them, Please teach

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

A generating function is nothing more than a power series where we do not care about the usual kaboobly doo called analysis, Radii of convergence, questions of convergence, analytical disks and even what the series converges to do not count.

What is unique about a gf is the coefficients are the number ways of some combinatorics problem. Now, that is very very loose but to show more I would need some examples.

I believe that you asked me a question similar to this one on another forum. It was about partitions, the major reason why Euler helped to invent gf's.

Here we have the gf of the Fibonacci sequence. Want to see how?

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

Combinatorics is Algebra and Algebra is Combinatorics.

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Yes. Please show

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

We start with the Daddy of all knowledge, the Taylor series.

If f(x) = x / ( 1- x - x^2) then we get:

Recognize those coefficients?

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

Combinatorics is Algebra and Algebra is Combinatorics.

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Yes.

I do not know much about the Taylor series though.

How did you expand that?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

That formula up there is the Taylor series. It says that most any function can be converted to and infinite series consisting of powers of x. ( some reservations here but trying to keep it simple and understandable)

For instance:

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

Combinatorics is Algebra and Algebra is Combinatorics.

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I entered the following into Maxima and it worked.

`taylor( x / ( 1- x - x^2), x, 0, 8);`

next...

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

Now you are learning!

When there are infinite number of terms we call them a Taylor series when we truncate at say 5 terms then we have a Taylor polynomial.

Now what is so useful about a polynomial instead of the function it replaces? To answer that kind of question you have to know the history of mathematics, something sadly lacking in today's syllabus.

Anyway, if we ever get into the uses of a Taylor series, I mean the real uses not 2 pages devoted to it in some calculus book, then questions of convergence and real analysis apply.

But we are getting away from gf's. You should see the Taylor series has as its coefficients the Fibonacci sequence and they are indexed correctly. We say that x / ( 1- x - x^2) is the generating function of the Fibonacci sequence.

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

Combinatorics is Algebra and Algebra is Combinatorics.

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We say that x / ( 1- x - x^2) is the generating function of the Fibonacci sequence.

So far, so good...

Now, how did this help you doing the problem?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

Did you understand what I meant when I said indexed correctly?

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

Combinatorics is Algebra and Algebra is Combinatorics.

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I overlooked that. Maybe you mean their order?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,383

That is correct.

Notice the first Fibonacci number is 1 and it is the coefficient of x^1. The fourth Fibonacci number is 3 and it is the coefficient of x^4 Likewise, the tenth is 55 and it is the coefficient of x^10.

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

Combinatorics is Algebra and Algebra is Combinatorics.

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