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#1 2013-11-19 09:01:05

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

Do you love limits?

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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#2 2013-11-19 09:27:55

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: Do you love limits?

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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#3 2013-11-19 10:50:13

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,525

Re: Do you love limits?

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)


“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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#4 2013-11-20 02:56:48

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

Re: Do you love limits?

Yes i am sorry at 2) was +...+
@anonimnystefy No it is f(c)=g(c)

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#5 2013-11-20 04:24:48

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: Do you love limits?

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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#6 2013-11-20 05:47:22

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

Re: Do you love limits?

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

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#7 2013-11-20 05:51:14

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: Do you love limits?

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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#8 2013-11-20 05:59:40

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

Re: Do you love limits?

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

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#9 2013-11-20 06:01:03

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: Do you love limits?

Constant what does that mean?

Infinity is not a constant.


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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#10 2013-11-20 06:12:15

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

Re: Do you love limits?

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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#11 2013-11-20 06:14:43

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: Do you love limits?

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


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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#12 2013-11-20 06:27:35

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

Re: Do you love limits?

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

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#13 2013-11-20 06:32:00

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: Do you love limits?

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.


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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#14 2013-11-20 06:37:56

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,525

Re: Do you love limits?

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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#15 2013-11-20 06:39:20

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

Re: Do you love limits?

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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#16 2013-11-20 06:39:50

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: Do you love limits?

How fast does the sum converge?

The sum as given does not converge.

Lagrange what?


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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#17 2013-11-20 06:42:35

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,525

Re: Do you love limits?

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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#18 2013-11-20 06:43:54

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: Do you love limits?

Slower than 1 / n that is for sure.


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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#19 2013-11-20 06:44:47

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

Re: Do you love limits?

Lagrange Theoreme for derivates

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#20 2013-11-20 06:46:14

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: Do you love limits?

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


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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#21 2013-11-20 06:46:50

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,525

Re: Do you love limits?

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.

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

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#22 2013-11-20 06:48:26

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

Re: Do you love limits?

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

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#23 2013-11-20 06:50:48

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: Do you love limits?

That is okay.

Are you sure that you have the right question?


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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#24 2013-11-20 06:51:57

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,525

Re: Do you love limits?

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.

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

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#25 2013-11-20 06:52:55

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: Do you love limits?

Hi;

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


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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