Do you love limits?

Yusuke00 · 2013-11-19 09:01:05

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)

bobbym · 2013-11-19 09:27:55

Hi Yusuke00

For 2)

anonimnystefy · 2013-11-19 10:50:13

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)

Yusuke00 · 2013-11-20 02:56:48

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

bobbym · 2013-11-20 04:24:48

Hi;

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

The integral test works well to prove divergence.

Yusuke00 · 2013-11-20 05:47:22

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

bobbym · 2013-11-20 05:51:14

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.

Yusuke00 · 2013-11-20 05:59:40

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

bobbym · 2013-11-20 06:01:03

Constant what does that mean?

Infinity is not a constant.

Yusuke00 · 2013-11-20 06:12:15

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.

bobbym · 2013-11-20 06:14:43

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

Yusuke00 · 2013-11-20 06:27:35

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

bobbym · 2013-11-20 06:32:00

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.

anonimnystefy · 2013-11-20 06:37:56

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)

Yusuke00 · 2013-11-20 06:39:20

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).

bobbym · 2013-11-20 06:39:50

How fast does the sum converge?

The sum as given does not converge.

Lagrange what?

anonimnystefy · 2013-11-20 06:42:35

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

bobbym · 2013-11-20 06:43:54

Slower than 1 / n that is for sure.

Yusuke00 · 2013-11-20 06:44:47

Lagrange Theoreme for derivates

bobbym · 2013-11-20 06:46:14

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

anonimnystefy · 2013-11-20 06:46:50

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.

Yusuke00 · 2013-11-20 06:48:26

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

bobbym · 2013-11-20 06:50:48

That is okay.

Are you sure that you have the right question?

anonimnystefy · 2013-11-20 06:51:57

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.

bobbym · 2013-11-20 06:52:55

Hi;

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

Math Is Fun Forum

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

Do you love limits?

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Re: Do you love limits?

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