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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Any other way? This one seems pretty impractical.

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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: 87,258

Hmmm, impractical? In 135 years of working with that rule it failed only once! Now that may not be good enough for a load of mathematicians like A that never calculated anything but it is a workhorse for us.

Sometimes we can do a tail analysis which can help us know how close we are. You are aware of the methods of numerical integration? Know some of the theory?

**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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

I'm not sure. I have heard of a few formulas, including the trapezoidal, simpsons, romberg,... Of these, I know only what the trapezoidal is. I am willing to learn it though.

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: 87,258

You know of a simpler one than those. You know about stuffing rectangles to get an area?

By the way, when you say things like you said in the previous post I am reminded of what pappym told me.

bobbym, everything you know is wrong! - pappym

http://www.youtube.com/watch?v=z4jeREy7Pbc

Everything they taught you is wrong. Think for yourself, do not opine the words of others. The double method is highly practical and we can improve 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.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Oh, yes, Riemann integration. I also remember Euler-Mclaurin...

So, what kind of tail analysis do you have in mind?

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: 87,258

The simplest one, If areas ( integrals ) can be approximated by rectangles then rectangles can be approximated by integrals. Each 1 x n rectangle can be thought of as a term in the sum.

Testing this we see that we are off by

by actual computation. There are some conclusions, notice that the double rule was predicting 8 and we really had about 10. Showing that it is on the conservative side and can be trusted.

**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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Ah, that one's nice.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,258

With it you can answer your question of how many terms you would need for 100 digits of accuracy.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

10^50 with the original series! That is really a lot!!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,258

For that many digits none of the methods looked at so far will suffice.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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I know. It needs exponential convergence or some trick, I guess...

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,258

We will get into a few sequence converters that might help tomorrow.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Can't wait. Tomorrow, then!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,258

I did a little preliminary work. It was not very encouraging.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

So, those sequence converters. What do they look like?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,258

I use a couple of them. Shanks, Romberg, Aitken, Euler, RRA, and a new one.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Well, how does the first one work?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Romberg acceleration worked best so far. We will start there.

Romberg integrates by taking a sequence of values and using Richardson extrapolation on them. This means it can accelerate many sequences. Have you ever programmed a Romberg integration?

If you have not and do not know how to write it I will give you mine.

See you a little later, chores are calling.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

I haven't programmed it, but I remember you helping me find it among the Maxima packages and use it for the 1/(cos x+x^2) integral.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,258

Hi;

I can give you mine if you want and we can go on.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

That would be okay.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,258

Hi;

```
romberg[l_] := Module[{x, a, o, l1}, a = Dimensions[l][[1]]; x = 1/2;
o = Table[0, {a}, {a - 1}];
l1 = augment[Transpose[{l}], o];
For[i = 2, i <= a, ++i,
For[j = 2, j <= a, ++j,
l1[[i, j]] = (l1[[i, j - 1]] -
x^(j - 1)*l1[[i - 1, j - 1]])/(1 - x^(j - 1));];];
l1[[a, a]]];
augment[a_, b_] := Transpose[Join[Transpose[a], Transpose[b]]];
```

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Does it work?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,258

Yes, try it on this simple one:

romberg[{1, 5/4, 205/144, 1077749/705600,

822968714749/519437318400}]

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Isn't it better to use recursion?

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