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#1 2011-07-30 01:13:40

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Series

Problems on partial / infinite series.

Prove that

1)


2)

Deduce that


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#2 2011-07-30 04:04:39

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi gAr;


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 2011-07-30 04:14:49

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#4 2011-07-30 04:19:08

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi gAr;

I know but I could not resist.


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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#5 2011-07-30 04:28:02

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series

Okay!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#6 2011-07-30 04:45:37

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi gAr;

Working on it the right way now, will post the answer if I get it.


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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#7 2011-07-30 04:50:05

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series

Hi,

Okay, take your time.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#8 2011-07-30 16:59:50

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi gAr;

Nothing yet, just some dead ends.


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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#9 2011-07-30 17:15:27

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series

Hi bobbym,

Okay,
a

Last edited by gAr (2011-07-30 17:23:58)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#10 2011-07-30 17:19:58

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi gAr;

I do not need the hint yet, have a couple of ideas left. Taking a break to rest. See you them.


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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#11 2011-07-30 17:22:32

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series

Okay, see you later.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#12 2011-07-30 21:55:10

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi gAr;




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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#13 2011-07-30 22:30:57

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series

Hi bobbym,


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#14 2011-07-30 22:38:22

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi gAr;


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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#15 2011-07-30 22:48:57

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series

Okay.

I was confused by W|A's output!

Isn't that always 0?


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#16 2011-07-30 23:00:03

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi gAr;

I do not think so. I solved for it as a definite integral:

integrate(log(2*cos(x/2))cos(n*x),{x,-Pi,Pi})/Pi

I just plugged in n = 0, 1, 2, 3, 4, 5

That works at home but Alpha is dogging it.


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 2011-07-30 23:07:34

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series

Hi,

Okay.
Could you deduce the sum?


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#18 2011-07-30 23:10:59

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi;

To get a feel for the coeficients:

nintegrate(log(2*cos(x/2))*cos(n*x),{x,-Pi,Pi})/Pi

Substitute for n =0,1,2,3,4,5 one at a time.

Could you deduce the sum?

If I have to use 1), so far I do not have any idea how to do it.


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 2011-07-30 23:16:46

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series

I have confirmed the coefficients.
In sage, numerical integral is okay, but integral shows 0. Even the sum, for some n, it goes completely wrong!


Do you need a hint or will you try some more time?


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#20 2011-07-30 23:33:12

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi gAr;

A hint? I need a doctor! I will try to do it as soon as I finish eating and take a little break.

Thanks for posing the problem.


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 2011-07-30 23:36:31

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series

Hi bobbym,

Okay, see you later.
I think I sounded like I was in a hurry, sorry.

You're welcome!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#22 2011-07-31 00:15:04

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi gAr;

No problem, do not worry about it.


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.

Offline

#23 2011-07-31 01:18:23

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series

3)

Find


Last edited by gAr (2011-08-25 22:48:55)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#24 2011-07-31 17:45:15

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Series

4)

This appeared in AMM

Find


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#25 2011-07-31 22:21:16

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,358

Re: Series

Hi;

I have been unable to deduce:

from

Did you have a solution?


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.

Offline

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