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#1876 2013-08-19 14:24:00

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

Hi;

A little more checking and it will be ready.


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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#1877 2013-08-19 19:14:57

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#1878 2013-08-19 23:13:29

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

Have you tested it?


“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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#1879 2013-08-19 23:20:56

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

Oh, and, a general formula for the line and squares problem:


“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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#1880 2013-08-20 00:01:49

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#1881 2013-09-27 07:24:00

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

bobbym wrote:

New Problem:

E says)

The sum

converges slowly. It takes 80,000 terms to get only 8 correct digits past the decimal. B can you speed up the convergence of this series? I know that Mathematica or Maple can get any number of digits for it but I would like to do it myself. What I am looking for is an acceleration of the series convergence.

A says) Hold on E, B isn't the only guy who can do things. I have never been wrong yet have I? Of course not. Just let me say there is no way. Now you can talk B.

B says) Thanks A you are too kind to me. Of course there is a simple way to produce a series that converges faster than that one. All you do is make use of...

C says) There you go again B. Contradicting A. There is a reason A goes first. He is the man! The best! He already said it can not be done.

D says) What is a series?

E says) A comes before B because of the dictionary. I want to hear what B has to say.

A says) I don't.

C says) Agreed!

D says) B, is it true that you are a distant cousin of Torricelli? Not Evangelista, the other one.

B says) Oh boy!?

Can you come up with a faster series that converges to the same value?

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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#1882 2013-09-27 08:14:46

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

Hi;


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.

Online

#1883 2013-09-27 08:19:45

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

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)


“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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#1884 2013-09-27 08:27:35

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#1885 2013-09-27 08:46:10

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

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.

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

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#1886 2013-09-27 08:49:32

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#1887 2013-09-27 08:51:46

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

Which trick?


“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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#1888 2013-09-27 08:53:11

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#1889 2013-09-27 09:06:15

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

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:


“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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#1890 2013-09-27 09:14:42

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#1891 2013-09-27 09:24:11

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

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?


“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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#1892 2013-09-27 09:28:35

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

Which sum do you want to do?


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.

Online

#1893 2013-09-27 09:29:59

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

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


“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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#1894 2013-09-27 09:33:51

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#1895 2013-09-27 09:36:43

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

That does not answer my question of how to actually get the number of needed terms...


“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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#1896 2013-09-27 09:43:04

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#1897 2013-09-27 09:52:03

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

But, in the original problem, you said we need 80000 terms. How did you get that number?


“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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#1898 2013-09-27 09:54:45

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#1899 2013-09-27 09:57:58

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Is this cool with you?

Give me something concrete, please. I still haven't the slightest how to get that estimate.


“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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#1900 2013-09-27 10:01:39

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,357

Re: Is this cool with you?

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

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