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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

Hi;

A little more checking and it will be ready.

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

Hi;

The formula is reasonably safe but who knows?!

The sum that generates the coefficients is:

Where

balls = total number of balls

urn = number of urns

max = maximum number of balls in any urn.

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,609

Have you tested it?

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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Oh, and, a general formula for the line and squares problem:

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Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

I have tested it a great deal it will produce the coefficients of the gf.

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,609

bobbym wrote:

New Problem:

E says)

The sum

I have this

, but I don't how much faster it is...Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

Hi;

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,609

Hi bobbym

I didn't know of any acceleration method besides the RRA, and that one works for alternating sequences only, so I used one I found on Wolfram MathWorld.

What do you have?

*Last edited by anonimnystefy (2013-09-27 08:19:58)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

There are many other methods besides RRA. In addition, any series can be converted into an alternating one and then RRA or Euler applied.

A much better approach is this one mentioned in the Scheid book.

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,609

Which is in turn

, so it's 1/k^5 rate of convergence!Here lies the reader who will never open this book. He is forever dead.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

That is true and the whole point but you had to see a trick first...

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,609

Which trick?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

The trick is how to sum that else you have replaced one cubic convergence with another. Do you see it?

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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I do. Partial fractions and telescoping. It is what the page I found used. http://mathworld.wolfram.com/Convergenc … ement.html

Also, I found this quintic convergence series:

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

So you are only left with the sum on the extreme right and it has much faster convergence. Now you should numerically verify that.

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,609

Hi bobbym

I cheated and checked with M that it does actually get the right answer.

What I do not know is how do we estimate how many terms are needed for some accuracy?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

Which sum do you want to do?

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,609

All mentioned so far, starting with the first one (sum of 1/k^3)...

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

Corny questions that come up in math courses which ask how many terms you need are replaced by what does the sum converge to and to how many digits can we get.

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

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**anonimnystefy****Real Member**- From: The Foundation
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That does not answer my question of how to actually get the number of needed terms...

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

I am not following you. It is a computational problem. To get the terms you have to use a computer and add them up. Then you need a tail analysis, remember most of the time you do not know what the sum is.

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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But, in the original problem, you said we need 80000 terms. How did you get that number?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

There are a couple of easy ways to back that statement up.First and simplest rule of thumb is the double rule.

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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Give me something concrete, please. I still haven't the slightest how to get that estimate.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

We can use the simplest command in M.

Sum[1/n^3, {n, 1, 40000}] // N

1.2020569028471022

Sum[1/n^3, {n, 1, 80000}] // N

1.2020569030814703

That is called the double method. Notice 8 digits passed the decimal point agree. You can expect the second answer is about accurate to 8 places.

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

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