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How do I approximate the following Triple Sum?

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'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,423

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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Never mind.

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

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,423

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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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 .**LearnMathsFree: Videos on various topics.New: Integration Problem | Adding FractionsPopular: Continued Fractions | Metric Spaces | Duality**

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,423

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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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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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
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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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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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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,423

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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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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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,423

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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Yep, that lower bound is correct.

How about 8 digits?

'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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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,423

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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Yes, I have a closed form answer

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,423

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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But how did you go about doing it?

'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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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
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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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And that gave you zeta(3)?

'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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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
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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.**

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