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MathGroup Archive 2000

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Re: Limit problem from analysis

  • To: mathgroup at smc.vnet.net
  • Subject: [mg21527] Re: [mg21495] Limit problem from analysis
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Fri, 14 Jan 2000 02:43:43 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Here is the more elementary (and longer)  proof I mentioned in my earlier
message.

Since we have 

In[46]:=
D[x^(1/x), x] // Simplify

Out[46]=
  -2 + 1/x
-x         (-1 + Log[x])

we see that the function x^(1/x) is decreasing for x>e.   Let's write
n^(1/n) = 1+ x[n], where x[n] is a decreasing sequence (for n>3) which is
bounded below by 0. We shall show that the limit of x[n] is exactly zero.
Suppose there is an A>0 such that x[n]>= A for all integers n. We have
n=(1+x[n])^n >= (1+A)^n for all n.
But it is esy to see that this can't be true for all n. (Just use the
binomial theorem to expand the right hand side and you can see that by
choosing n large enough we can make the right hand side >n ).
So there can't be such an A. Therefor the greatest lower bound for the
sequence x[n] is 0, hence lim x[n] = 0 hence lim n^(1/n) = 1.

> From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
To: mathgroup at smc.vnet.net
> Date: Thu, 13 Jan 2000 19:06:21 +0900
> To: <world at writemaster.com.xxx>, <mathgroup at smc.vnet.net>
> Subject: [mg21527] Re: [mg21495] Limit problem from analysis
> 
> Here is a very short proof, which I think satisifes your conditions (
> depending of course on what one thinks should be contained in the first few
> chapters of an analysis book):
> 
> To show that lim x^(1/x) as x goes to infinity is 1 is equivalent to showing
> that lim log(x^(1/x)) as x goes to infinity is 0. This in turn amounts to
> showing that log(x)/x goes to 0 as x goes to infinity. But this follows from
> the well known L'Hospital rule.
> 
> I am sure that a more elementary proof can be given but it is likely to be
> rather longer.
> 
> 
>> From: world at writemaster.com.xxx
To: mathgroup at smc.vnet.net
>> Organization: Concentric Internet Services
>> Date: Wed, 12 Jan 2000 08:35:39 -0500 (EST)
>> To: mathgroup at smc.vnet.net
>> Subject: [mg21527] [mg21495] Limit problem from analysis
>> 
>> I am teaching myself analysis, and I'm stuck on a particular problem.
>> The problem is to prove that the limit as n goes to infinity of n to
>> the (1/n) power is 1.  The only allowed tools are those of the first
>> few chapters of a basic analysis textbook.  The hint given in the book
>> is to use the binomial theorem.
>> 
>> You can find the work I've done at
>> http://writemaster.com/public/mathPostings/
>> There are two documents there, one a Mathematica notebook and the
>> other an MS Word document.   Any suggestions for how to proceed, or
>> even outright solutions, would be welcome.
>> 
>> I am not taking any classes, and you will not be helping me with a
>> test or homework.
>> 
>> Steve Oppenheimer
>> writer at writemaster.com
>> 
>> 
>> 
>> 
> 



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