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#1 2013-08-10 10:24:14

mukesh
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limit

evaluate
lim n!/n^n
n->infinity

#2 2013-08-10 10:34:09

ShivamS
Super Member

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Re: limit

0.


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#3 2013-08-10 10:45:21

mukesh
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Re: limit

hw?i didnt undrstand

#4 2013-08-10 21:30:08

bobbym
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Re: limit

Intuitively obvious that it approaches 0.


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 2013-08-10 22:02:09

bob bundy
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Re: limit



(capital pi symbol is for 'the product of' )

i/n ≤1 => the product is less than 1

as n approaches infinity 1/n approaches zero = the limit tends to zero

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#6 2013-08-17 01:58:41

anonimnystefy
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Re: limit

Hi Bob

Is there somethingissing from the explanation?


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

#7 2013-08-17 02:15:06

bob bundy
Moderator

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Re: limit

hi Stefy

Is there somethingissing from the explanation?

aybe.  y posts often have things issing.  But what would you like e to include?

I'd put in an 'm' if I could think of a suitable place for it.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#8 2013-08-17 05:00:04

anonimnystefy
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Re: limit

Typing issues, sorry.

The last sentence of your explanation doesn't follow from the rest.


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

#9 2013-08-17 05:27:29

SteveB
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Re: limit

Maybe you should use the fact that (1/n) is one of the terms of the product and all of the others are one or less.
Then use the fact that if you multiply by a number between zero and one it reduces a positive number or keeps
it the same when it is equal to one.
Then I think you can make Bob's deduction.
In a formal proof that needs to be written out correctly. If you are doing a pure maths related course, then I
should do the rest as an exercise, mukesh, if you just copy someone else's version then you won't understand it.
anonimystefy: I agree. Strictly speaking in a formal proof that probably isn't enough. Suppose the product sequence
converges to one with increasing density, as n inreases, all near to one. The product could converge to a higher number.

Last edited by SteveB (2013-08-17 05:29:12)

#10 2013-08-17 05:46:32

bobbym
Administrator

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Re: limit

Hi;


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.

#11 2013-08-17 07:46:01

bob bundy
Moderator

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Re: limit

I have shown that the product can be written



As 1/n tends to zero as n tends to infinity, I claim the product tends to zero times something finite and therefore to zero.

What's wrong with that?

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#12 2013-08-17 17:09:45

SteveB
Power Member

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Re: limit

I agree with Bob. My only criticism was a lack of clarity of the argument. I knew what you meant and it is okay as
a proof provided:
(1) The result concerning (1/n) is proven earlier in the course and can be used by referring to it (it should be usually).
(2) The fraction has to be trapped below a certain positive constant. In this case it is obviously one or less.
Each term is between zero and one, including a term equaling one.
Proving bobbym's identities looks difficult to me, so if the question is an exercise in proving things then they might be
challenges to go on to if the proof of the first problem was too easy.

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