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**Yusuke00****Member**- Registered: 2013-11-19
- Posts: 43

Hey guys,i'm new here. Hope u can help me with those 3 problems:

1)Let it be f,g:[-1,1]-> R; f,g - continous functions.Show that exist a,b ∈[-1,1],a<b so that f(a)=g(b) and f(b)=g(a) then there is c∈[-1,1],so that f(c)=g(c).

My guess here it's Lagrange but i have no clue on how to apply it.

2)Calculate lim n->inf ( 1/(2ln2)+1/(3ln3)+...+1/(nln(n)) )

3)a,b,c > 0 so that a^x+b^x+c^x>=3, with any x∈R. Show that a*b*c=1

Ty for help xD

*Last edited by Yusuke00 (2013-11-20 02:50:41)*

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

Hi Yusuke00

For 2)

**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: 15,522

I think he might have meant 1/(2log2)+1/(3log3)+...+1/(nlogn).

For the first problem, did you mean f'(c)=g'(c)?

*Last edited by anonimnystefy (2013-11-19 10:55:11)*

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

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**Yusuke00****Member**- Registered: 2013-11-19
- Posts: 43

Yes i am sorry at 2) was +...+

@anonimnystefy No it is f(c)=g(c)

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

Hi;

Have you tried some tests on 2) which is really only:

The integral test works well to prove divergence.

**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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**Yusuke00****Member**- Registered: 2013-11-19
- Posts: 43

They don't ask to prove the divergence but a result.

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

If it diverges it means the sum is infinity. That is what I am talking about. Use the integral test and you will prove the sum diverges.

**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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**Yusuke00****Member**- Registered: 2013-11-19
- Posts: 43

Actually the sum is constant,that's a thing i know.

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

Constant what does that mean?

Infinity is not a constant.

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

Combinatorics is Algebra and Algebra is Combinatorics.

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**Yusuke00****Member**- Registered: 2013-11-19
- Posts: 43

it's 7e/16 i think.

ok the thing is that i did this exercise 2 years ago but i lost the notebook and now i can't figure out how to solve it anymore

i know that the sum is constant and i know it's something with xe/12 or xe/12 but i cannot remember exactly.

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

Why do you think that sum converges besides from your memory?

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

Combinatorics is Algebra and Algebra is Combinatorics.

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**Yusuke00****Member**- Registered: 2013-11-19
- Posts: 43

I am sorry you were write,i just found the notebook.apologize

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

Hunches are fine, but in math you will have to back them up with numerical evidence or proof.

Using the integral test:

http://en.wikipedia.org/wiki/Integral_t … onvergence

If that integral converges then so does the sum, if it diverges so does the sum:

Say u = log(x), then du/dx = 1 / u, and the integral becomes

Substituting back u = log(x)

log(log(x)). Taking the limits of integration.

The integral is infinity, so it diverges. The sum also equals infinity, so it diverges.

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: 15,522

The way the question is posed (with the limit instead of infinity as the upper sum limit), I would guess there is a missing term in front of the sum.

Hi bobbym

How fast does the sum converge?

*Last edited by anonimnystefy (2013-11-20 06:38:17)*

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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**Yusuke00****Member**- Registered: 2013-11-19
- Posts: 43

Yes that's a part i don't understand yet.I'm on last year of high school so i started learning about integrals just now.I still have to learn a bit more about integrals because at the time i solved it i used just Lagrange and derivates.

I also found out how to solve 3) if you are interested but still no clue for 1).

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

How fast does the sum converge?

The sum as given does not converge.

Lagrange what?

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: 15,522

Not converge, sorry. What's its growth rate?

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

Slower than 1 / n that is for sure.

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

Combinatorics is Algebra and Algebra is Combinatorics.

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**Yusuke00****Member**- Registered: 2013-11-19
- Posts: 43

Lagrange Theoreme for derivates

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

Lagrange probably has 2000 theorems named after him. Do you mean the mean value theorem?

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: 15,522

bobbym wrote:

Slower than 1 / n that is for sure.

I'm thinking it's O(log log n). Can you do the same limit, but with an 1/n in front of the sum.

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

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**Yusuke00****Member**- Registered: 2013-11-19
- Posts: 43

Yes the mean value theorem.Sorry they don't call like that here.

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

That is okay.

Are you sure that you have the right question?

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: 15,522

Hi bobbym

anonimnystefy wrote:

bobbym wrote:Slower than 1 / n that is for sure.

I'm thinking it's O(log log n). Can you do the same limit, but with an 1/n in front of the sum.

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

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

Hi;

Yes, I saw that. Why are we doing that limit?

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

Combinatorics is Algebra and Algebra is Combinatorics.

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