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Yes, that I've seen

'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'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

Now the notation does the rest. We can write

Where

is the nth Fibonacci number.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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Hmmm...

The next challenge is to find something whose taylor series is 5x + 5^2*x^2 + 5^3*x^3....

Which we could do with the formula of GP but to bad, I've forgotten 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'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

One second we are getting there.

We can further write.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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Yes, thats just the same thing with some more notation

'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'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

Okay, if we substitute 1 / 5 for x we get, (1/5), ( 1/5)^2, (1/5)^3...

or written as

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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yes great! Awesome!

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

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

I have not changed anything on the LHS, so lets do that now.

Make the substitution:

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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We are done with that, I guess...

Now the big question is how did you find a function whose taylor series contains the fibonacci?

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

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

You do not have to, x / (1 - x - x^2) is the generating function. Each problem has a different gf usually. Some are so useful they have hundreds of uses for that gf like the Catalan numbers,

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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x / (1 - x - x^2) is the generating function.

How did you come to know?

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

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

Because basically they are derived from the recurrence that states the problem. f(n) = f(n-1) + f(n-2). That is one way.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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Would you show their derivation?

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

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

Hi;

Off hand I do not remember how. I will post when I find it.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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http://en.wikipedia.org/wiki/Fibonacci_ … wer_series

Please see this and explain it to me

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

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

Hi Agnishom;

I do not like his proof. For one thing, F0 makes little sense to me.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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But, I thought you were talking about this one

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

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

Hi;

Straight out of Wilf's book.

We start with:

multiply the LHS by x^n and sum over n+1 to get,.

On the RHS we get:

Solving for F(x) we get:

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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How did you get that by multiplying the LHS with x^n?

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

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

Beats me! Herbert is about 200 times smarter than me.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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Does it mean you understand it neither?

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

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

I do not understand his derivation. I rarely understand someone else's proof or program. I always have to do it myself to have any chance.

I had a method that was so good it worked on a small programmable calculator. I do not remember it though.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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What is a programmable calculator?

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

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,585

Today they call them graphic calculators.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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How is the taylor series of sin found out?

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

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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