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#1 2013-03-30 12:31:44

White_Owl
Member
Registered: 2010-03-03
Posts: 99

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

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

My Answer:
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...

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#2 2013-03-30 13:10:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,303

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

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#3 2013-03-30 13:54:10

White_Owl
Member
Registered: 2010-03-03
Posts: 99

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

I do not think there are mistakes:


Or are you talking about different equations?

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#4 2013-03-30 14:02:41

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,303

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

Your first line is good.

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

Online

#5 2013-03-30 14:09:23

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

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))


“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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#6 2013-03-30 14:14:01

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,303

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

Online

#7 2013-03-30 14:17:31

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

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

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


“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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#8 2013-03-31 11:28:04

White_Owl
Member
Registered: 2010-03-03
Posts: 99

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

anonimnystefy, thank you. I see the mistake now

So my new answer is:


Therefore, series converges for k>=2

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#9 2013-03-31 11:32:10

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

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".


“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 2013-03-31 15:39:54

White_Owl
Member
Registered: 2010-03-03
Posts: 99

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

Yes, of course. Thank you.

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#11 2013-03-31 15:44:27

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

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

Everything else seems okay to me.

You are welcome.


“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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