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MathGroup Archive 2000

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Re: Limit of sequence

  • To: mathgroup at smc.vnet.net
  • Subject: [mg25290] Re: [mg25276] Limit of sequence
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Tue, 19 Sep 2000 03:45:13 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

on 9/18/00 6:33 AM, Kawassaki at none at none.com wrote:

> Hello,
> 
> Can Mathematica find the limit of a sequence?
> As I understood, it treats the argument of the Limit function as a real
> variable,
> therefore the limit of Sin[Pi*n] turns out to be undefined.
> 
> Thanks,
> Nadav.
> 
> 
> 
> 

This is true, but there are some ways around it. This particular case is
trivial and there are at least two ways in which Mathematica can "do" it:

In[4]:=
Limit[Simplify[Sin[Pi*n], Element[n, Integers]], n -> Infinity]
Out[4]=
0

This is of course "cheating" since Limit has essentially nothing to do here.
A somehwat more convincing way is:

In[5]:=
Sum[Sin[(n + 1)Pi] - Sin[n*Pi], {n, 1, Infinity}] + Sin[Pi]
Out[5]=
0

This indeed does compute the limit of the sequence adn can in fact be used
to find some not entirely trivial limits. However, for the really
interesting ones you have to use your own brain. For example, I can't think
of any way to use Mathematica to show that the limit of the sequence
a[n_]:=n*Sin[2Pi*E*n!] is 2Pi.
-- 
Andrzej Kozlowski
Toyama International University, JAPAN

For Mathematica related links and resources try:
<http://www.sstreams.com/Mathematica/>



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