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#1 2013-03-31 11:31:44

White_Owl
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Convergence of (n!)^2/(kn)!

The problem is:
For which positive integers k is the following series convergent?

For series to be convergent the next inequality should be true (by the Ratio Test):

Since we know that both k and n are positive we can omit absolute bars.

And now I simplify:

But since k is a constant this limit will never be less than 1. Therefore the series divergent for all possible k.

Did I make a mistake somewhere? Textbook is looking for a convergent series...

#2 2013-03-31 12:10:38

bobbym

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Re: Convergence of (n!)^2/(kn)!

Hi;

Something is wrong right there.

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.

#3 2013-03-31 12:54:10

White_Owl
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Re: Convergence of (n!)^2/(kn)!

I do not think there are mistakes:

Or are you talking about different equations?

#4 2013-03-31 13:02:41

bobbym

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Re: Convergence of (n!)^2/(kn)!

Shouldn't manipulations maintain equality with the original assertion?

That does 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.

#5 2013-03-31 13:09:23

anonimnystefy
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Re: Convergence of (n!)^2/(kn)!

White_Owl wrote:

I do not think there are mistakes:

Or are you talking about different equations?

The second line is not correc(k(n+1))!=1*2*3*...*(k(n+1)-1)*(k(n+1))

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#6 2013-03-31 13:14:01

bobbym

Online

Re: Convergence of (n!)^2/(kn)!

Hi;

That is what I mean, something is bad where I indicated. There could be further mistakes but that is where the first one occurs.

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.

#7 2013-03-31 13:17:31

anonimnystefy
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Re: Convergence of (n!)^2/(kn)!

Well, the rest of his current work is okay. But that error is messing up the whole thing.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#8 2013-04-01 10:28:04

White_Owl
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Re: Convergence of (n!)^2/(kn)!

anonimnystefy, thank you. I see the mistake now

Therefore, series converges for k>=2

#9 2013-04-01 10:32:10

anonimnystefy
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Re: Convergence of (n!)^2/(kn)!

Third line - in the denominator you have k1*k2*...*kn and you say below it "n times". It should be 1*2*...*kn and below it should be "kn times".

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#10 2013-04-01 14:39:54

White_Owl
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Re: Convergence of (n!)^2/(kn)!

Yes, of course. Thank you.

#11 2013-04-01 14:44:27

anonimnystefy
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Re: Convergence of (n!)^2/(kn)!

Everything else seems okay to me.

You are welcome.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment