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#1 2012-04-21 09:59:50

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,420

A tough sum - limit?

The above problem came up in the now famous gAr series thread. Generally I stay away from that thread. It is for the members to work on and enjoy without me butting in.

I was asked for some help on this one and because this one has many interesting points I agreed. Of course before I attempt to answer I must do some ranting.

begin Rant():

I believe that this sort of problem is easily handled through the methods I have been going on and on about in this thread for more than a century now. Apparently I can foam at the mouth for as long as I like about how no one is reading any of it.

Why is that? Why is there not 1 million replies to each of these problems? Why do posters keep heading over to other forums to read non solutions full of homeomorphisms, iosomorphisms and more rings than a jeweler? Beats me!

End Rant:
Return(answer)

Here is how we can do it using the methods outlined here. These methods allow one to get exact answers to dificult problems using commonsense reasoning.

First thing we observe that as n gets larger so does k and so does k*n. That means those 3 constants a,b,c are going to be drowned out. We can reduce the problem to

and then to this.

This is easily handled by a CAS:

n = 300000000000000000000000;
NSum[1/Sqrt[k n + n^2], {k, 1, n}, WorkingPrecision -> 25]

the output is 0.8284271247461900976033770...

To show we are on the right track we do a little bit of experimenting. We choose three arbitrary values for a,b and c.

NSum[Sqrt[k n + n^2 + 31]/(
 Sqrt[k n + n^2 + 2] Sqrt[k n + n^2 + 5]), {k, 1, n}, 
 WorkingPrecision -> 25]

the output is 0.8284271247461900976033770...

We will get this for any a,b and c we choose.

Okay we have experimentally

what now? A PSLQ of course! We use one on the above constant and come up with:

We have a conjecture, a good one. We are done.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#2 2012-04-21 23:41:02

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: A tough sum - limit?

So,you hqve no definite proof of your answer.Ahhhhh... Well this will do I guess. But I would still like to do it by a hand method.


“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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#3 2012-04-21 23:43:32

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,420

Re: A tough sum - limit?

Does a hand method automatically mean a proof?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#4 2012-04-21 23:48:57

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: A tough sum - limit?

If it is correct,yes.


“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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#5 2012-04-21 23:49:52

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,420

Re: A tough sum - limit?

If it is long and difficult how do we verify its correctness?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#6 2012-04-21 23:51:40

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: A tough sum - limit?

How do we verify the correctness of your answer?


“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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#7 2012-04-21 23:55:52

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,420

Re: A tough sum - limit?

We are sure of the amount of digits -1 that are provided. It is now possible to back engineeer that result. One method suggests itself right away. The point is, it is much better to have 25 digit conjecture than a 0 digit nothing.

But it is okay that you want to do it your way. I can now submit my answer to the series thread. When you get yours you can submit that also.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#8 2012-04-21 23:59:21

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: A tough sum - limit?

Ok. I think I am gonna try it "my way",but even then I will use your result from computer math to help me. So it is not a waste in the end.

Last edited by anonimnystefy (2012-04-21 23:59:40)


“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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#9 2012-04-22 00:12:57

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: A tough sum - limit?

Do you have a suggestion for a hand method?


“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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#10 2012-04-22 00:21:20

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,420

Re: A tough sum - limit?

I am sorry, I do not.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#11 2012-04-22 09:01:07

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: A tough sum - limit?

Okay. Thank,anyways,for this method here.


“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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#12 2012-04-22 09:03:44

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,420

Re: A tough sum - limit?

I think my earlier comments about a gAr problem apply here doubly.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

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