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#1 2016-11-19 04:11:11

Agnishom
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From: Riemann Sphere
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Triple Summation

How do I approximate the following Triple Sum?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#2 2016-11-19 10:17:33

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 107,190

Re: Triple Summation

Hi;

Agnishom: I have a lower bound of 2.34 but nothing more than that.

zetafunc: I am getting about 1.64 for that double sum. Are you sure about 3?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2016-11-19 10:33:21

zetafunc
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Re: Triple Summation

Never mind.

Last edited by zetafunc (2016-11-19 10:44:51)

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#4 2016-11-19 10:41:20

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 107,190

Re: Triple Summation

I do not know the first line. What did you use there?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2016-11-19 10:45:55

zetafunc
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Registered: 2014-05-21
Posts: 1,735
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Re: Triple Summation

Sorry, I made a mistake. I mistakenly thought that

where
is the beta function, and
is the gamma function. But of course, the denominator is actually
not
.

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#6 2016-11-19 10:46:46

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 107,190

Re: Triple Summation

Does that change your answer?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2016-11-19 10:49:29

zetafunc
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Registered: 2014-05-21
Posts: 1,735
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Re: Triple Summation

Yes, because there will be an extra factor. You can however say that the double sum is:

then maybe replace the beta function with its integral representation and go from there. For Agnishom's triple sum, my idea is to either use a modified beta function (with 3 arguments rather than 2) and repeat the same procedure, or try to compare with the trinomial theorem.

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#8 2016-11-19 10:58:22

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 107,190

Re: Triple Summation

We are going to need some analytical approach for the 3 sum because numerically it is very difficult.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#9 2016-11-19 17:42:38

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,442
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Re: Triple Summation

How would an experimental mathematician do it?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#10 2016-11-19 19:11:40

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 107,190

Re: Triple Summation

How would an experimental mathematician do it?

He would try to get a very accurate numerical answer is one way. I have already given you a lower bound.

zetafunc: I would guess that your sum


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#11 2016-11-20 05:08:53

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,442
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Re: Triple Summation

That upper bound is correct. How do I do better?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#12 2016-11-20 06:49:38

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 107,190

Re: Triple Summation

How much accuracy do you require and that is a lower bound.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#13 2016-11-20 18:21:26

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,442
Website

Re: Triple Summation

Yep, that lower bound is correct.

How about 8 digits?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#14 2016-11-20 18:42:19

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,190

Re: Triple Summation

Yep, that lower bound is correct.

You have the answer already?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#15 2016-12-01 16:28:10

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,442
Website

Re: Triple Summation

Yes, I have a closed form answer


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#16 2016-12-01 19:26:11

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,190

Re: Triple Summation

Hmmm, I will try for a digit or two more but that will be difficult.

My closed form involves zeta(3).


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#17 2016-12-02 16:05:14

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,442
Website

Re: Triple Summation

But how did you go about doing it?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#18 2016-12-02 18:51:49

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,190

Re: Triple Summation

I used M of course and lots of CPU cycles.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#19 2016-12-03 13:22:53

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,442
Website

Re: Triple Summation

And that gave you zeta(3)?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#20 2016-12-03 13:49:57

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,190

Re: Triple Summation

No, there was more than that but zeta(3) was part of the conjectured form.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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