Limit of (-1)^n?

Date: 03/14/98 at 09:43:36
From: ANDRE BOURQUE
Subject: Need help please

Hi, 

My name is Andre, and I am a 3rd year student in electrical 
engineering. I am stumped on a little equation and would be grateful 
for your help.

The limit for n = infinity, for (-1)^n = ?

I am testing if the lim n->inf. ((-1)^n * (n^2)/(1+n^2))...
Thank you once more (you helped me before Christmas '97).

Andre Py Bourque

Date: 03/16/98 at 10:01:14
From: Doctor Timothy
Subject: Re: Need help please

Hey Andre,

I'd be glad to help. Limits can often be confusing to me, too. To 
answer your first question, let's look at

             n
     Lim  (-1)
    n-->inf.

Probably the easiest way to understand what's going on is to just 
start writing terms. Nothing too wild is happening:

     -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1,      
     -1, 1, -1, 1, -1, 1, -1, 1, -1... ENOUGH!

If you don't intuitively see the answer, let's look at things in a 
more mathematically rigorous way, by using our definition of limits. 
If the sequence _converges_ to some value, call it x, or if it has a 
limit of x, what that means is that if you pick a _neighborhood_ of 
ANY size around x, you can also find an integer "N" such that all 
elements of the sequence lie "inside" that neighborhood for n > N.

Hmmmm... let's try picking a limit of x = 1. I want you to find an N
such that every element of the sequence lies inside the set
(1/2, 3/2), called a "neighborhood of radius 1/2 about x = 1," 
for all n > N. (Note: don't try too hard; it's impossible! Can you see 
why?) I think you'll have the same problem for any limit point you try 
to pick.

Now that you understand the first problem, the second problem should
be much easier, right?

Thanks for writing! Feel free to write back if you have any more
questions or if anything is unclear.

-Doctor Timothy,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/