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#1 2016-12-13 21:39:42

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Because he asked.

This appears in another post.

There was some preliminary sparring done with the problem but it remained steadfast. It is no big secret that M's Sum and NSum are not going to work here.

A wrote:

How would an experimental mathematician do it?

I prefer "How would an EM guy do it?"  I do not believe in my heart that there is such a thing as an experimental mathematician or a truly experimental physicist anymore. Plato used to preach that the best way to do astronomy was to go in a cave and just think about it for awhile. Do not bother to look up at the sky. I think all mathematicians and physicists and a great deal of math types on forums secretely admire this approach greatly.

Here is one way that we can coax an answer out of M.

First define a function h

h[n_]:=Sum[N[(i!j!k!)/(i+j+k+2)!],{i,0,n},{j,0,\[Infinity]},{k,0,\[Infinity]}]//N

Form this sequence.

{h[200],h[400],h[800],h[1600],h[3200]}

That will take some time to do.

Now we are going to play "spot the pattern." Only, we are going to call forth the awesome power of M to help us.

The above sequence numerically looks like this

A)

{2.4624260169406718`, 2.464907339855726`, 2.466152661470262`, 2.4667764907345844`, 2.4670886979082494`}

Looks ugly does it not? Nope, it does not because to an EM guy that sequence is really singing like the lovely Emmy Rossum.https://www.youtube.com/watch?v=pgz6PnHkmpY

We apply a sequence accelerator because we know it will work well here. How do we know that? That is another post or two some other time. I like Shanks for acceleration sequences like these but maybe Romberg, Levin or Aitken might get there. Anyway,

shanks[{2.4624260169406718`, 2.464907339855726`, 2.466152661470262`, 2.4667764907345844`, 2.4670886979082494`}]

yields

B)

{2.4674073736800355`, 2.4674026656998325`, 2.4674014912640865`}

We can expect the last one 2.4674014912640865`, to be close. Now what we expected all along when we saw sequence A) was that we were dealing with

We can see we have an agreement of 6 digits after the decimal point.

A wrote:

How about 8 digits?

The last question:

A wrote:

How do I approximate the following Triple Sum?

Lots of times and this is becoming a habit, EM can get a closed form. We do not have to settle for an approximation. And, we are done.


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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#2 2016-12-14 00:28:19

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

There is a shanks command?


'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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#3 2016-12-14 04:00:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

Nope, I had to implement it. M forces you to learn more math so that you can converse with him intelligently.

Built in commands always are capitalized, I always use small letters to start mine.


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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#4 2016-12-14 13:26:35

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

Please tell me what shanks is?


'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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#5 2016-12-14 13:28:28

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

shanks is a sequence accelerator. If a sequence is converging in a specific way, the true convergent can be got from the terms you already have. You have seen this with series also.


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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#6 2016-12-14 13:37:08

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

Not really. If you do not tell me what shanks is, the rest of the post is not going to make sense to me


'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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#7 2016-12-14 13:39:15

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

Have you not been given the code for shanks accelerators?


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

#8 2016-12-14 13:43:33

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

I have not openned the video, if that is what you are talking about.


'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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#9 2016-12-14 13:46:11

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

The video is not about shanks.

Use this with all rights reserved.

shanks[l_] := Module[{x, n, w}, w = l; x = 1; 
    For[n = 3, n <= Dimensions[l][[1]], ++n, 
     w[[x]] = (w[[n - 2]]*w[[n]] - w[[n - 1]]^2)/
        (w[[n]] - 2*w[[n - 1]] + w[[n - 2]]); 
      x = x + 1; ];
 w = Take[w, Dimensions[w][[1]] - 2]
];

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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#10 2016-12-14 13:56:47

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

Now how did you get to pi^2/4?


'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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#11 2016-12-14 14:00:54

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

Hmmm, when I had h[1600] I got the idea, and FindInstance seemed to verify it. Of course, when I saw the form because of RIPOSTP, it all clicked together. I just needed one more to verify it. h[3200] did that.


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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#12 2016-12-14 14:02:40

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

How did you get that idea when you had h[1600]?


'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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#13 2016-12-14 14:05:41

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

When you have a number and wish to know the closed form for that number you have several options,

1) FindInstance is the most powerful command ever devised.
2) A PSLQ - called the greatest algorithm of the twentieth century it will transform mathematics. Axiomatic math -> EM


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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#14 2016-12-14 14:11:56

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

Okay. How did you use FindInstance here?


'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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#15 2016-12-14 14:13:27

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

FindInstance[{Abs[\[Pi]^a/b - 2.4667764907345844`] < .001, 10 > a > 0,
   b > 0}, {a, b}, Integers]

But that was not good enough, so I needed h[3200] and then shanks to verify the conjecture.


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

#16 2016-12-14 14:15:30

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

How did you know Pi is going to be involved?


'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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#17 2016-12-14 14:19:24

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

The All-Father wove the skein of your life a long time ago.

All things are connected, everywhere, everytime, everyone, all one. I was tearing up the 400 point problems at brilliant when I came across yours and when I solved it, I solved two problems at once. RIPOSTP.

It is important to know that even if I had not seen it there, I still had a very good chance to solve yours.


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

#18 2016-12-14 14:24:42

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

I know. Why did you post this in a new thread?


'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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#19 2016-12-14 14:29:12

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

Agnishom wrote:

I know.

It was not through any brilliance on my part. A PSLQ would have solved it. My main message has become to show the members of the forums I inhabit that even a bumpkin can solve the toughest problems if he will use EM.

I felt it had enough to be a standalone thread and I did not want to connect the problem here to the one there.

Also, One of the tenets of Baroque.

Eccentricity, - he walked on his hands to spare his feet.

Go out from amongst them and be ye separate.

Be gem, don't be wheat. - pappym

Hang out with people smarter than yourself. - grandpappyd

.
.
.


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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#20 2016-12-14 14:37:01

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

It is the same problem!


'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.

Offline

#21 2016-12-14 14:38:48

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

Yes, it is the same.

Agnishom wrote:

I know.

It was not through any brilliance on my part. A PSLQ would have solved it too.

Do you understand this process now?


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

#22 2016-12-14 21:28:46

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

Not really.

Can you give me a problem like this and walk me through?


'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.

Offline

#23 2016-12-15 02:14:45

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

Hmmm, but first I must rant:

Begin Rant:

Do you know that while math types know some methods to integrate and do that all day long, very few know anything about how to do a summation problem. Most have never heard of the summation calculus. What is strange to me is why do you have to learn two different methods when EM does both in just the same way? The Teakettle Principle strikes again!

End Rant:

Try your blaster there, commander - Edward Morbius.

Try this one...

and you know the answer is of the form a + b*ln(c) and a,b and c are positive integers. Or, if that one seems too daunting then try this instead:


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

#24 2016-12-17 01:05:48

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

Re: Because he asked.

bobbym wrote:

Most have never heard of the summation calculus.

Knuth talks about it in Concrete Mathematics!

For the sum,

So,

When we multiply both sides by (-1), we get the famous alternating harmonic series which we know to be ln(2):

Now, of course we should do some checking to see that our answers are making sense because

bobbym wrote:

EM guys always check their work!

Prelude> sum $ take 4096 [cos(pi*n)/n | n <- [1.0, 2.0 ..]]
-0.6930251251486118
Prelude> - (log 2)
-0.6931471805599453

Last edited by Agnishom (2016-12-17 01:09:56)


'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.

Offline

#25 2016-12-17 04:15:03

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Because he asked.

Can you give me a problem like this and walk me through?

Hmmm, you ask for an example to learn the technique. That is not the technique...


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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