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

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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
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From: Bumpkinland
Registered: 2009-04-12
Posts: 88,985

Re: Series

Hi gAr;


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

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#3 2011-07-30 04:14:49

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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: 88,985

Re: Series

Hi gAr;

I know but I could not resist.


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

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

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

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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."

Offline

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,985

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

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

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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."

Offline

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,985

Re: Series

Hi gAr;

Nothing yet, just some dead ends.


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

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

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

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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."

Offline

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,985

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

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

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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: 88,985

Re: Series

Hi gAr;




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

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

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

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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: 88,985

Re: Series

Hi gAr;


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

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

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

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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."

Offline

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,985

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#17 2011-07-30 23:07:34

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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: 88,985

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#19 2011-07-30 23:16:46

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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: 88,985

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#21 2011-07-30 23:36:31

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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: 88,985

Re: Series

Hi gAr;

No problem, do not worry about it.


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

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

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

gAr
Member
Registered: 2011-01-09
Posts: 3,479

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: 3,479

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: 88,985

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

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