Infinite Sequence

Date: 09/28/97 at 23:32:19
From: Moni
Subject: Calculus - infinite series

I'm having trouble with this question:

From f:n ---> 3 + (-1/2)^n where n belongs to the set of natural 
numbers

Describe a strip that contains all but a finite number of points
of the graph of f

I know that you first have to find the limit, which I think is 3,
and then choose E = 0.001

2 cases:

L + E = 3 + 0.001  and   L - E = 3 - 0.001
f(n) > 3.001       and   f(n) < 2.999

But I don't know what to do now.

Thanks.

Date: 09/29/97 at 11:19:30
From: Doctor Rob
Subject: Re: Calculus - infinite series

Point 1:  This is an infinite *sequence*, not a *series*.  An infinite
          series is the sum of the terms of an infinite sequence.

Point 2:  The graph of f is the set of pairs {(n, f(n)): n is a 
          natural number}.  A strip of this graph is defined by 
          inequalities involving n and/or f(n).

You have made a good start. Yes, the limit L = 3, and you have chosen
E = 0.001, which is fine. Your inequalities, however, are backwards, 
I think, since all but a finite number of values of f satisfy

   2.999 < f(n) < 3.001

These inequalities define a strip in the plane extending to infinity 
in both positive and negative n directions which contains all but a 
finite number of the points of the graph of f. If you mean the strip 
to extend only to infinity in the positive n direction, add the 
condition n >= 1.

Perhaps you meant f(n) > 3.001 and f(n) < 2.999 to be the start of the
process of finding excluded values of n.  If so, that is the way to 
start:

f(n) = 3 + (-1/2)^n > 3.001
 <==> (-1/2)^n > 0.001
 <==> n is even and (1/4)^(n/2) > 0.001
 <==> (n/2)*log(1/4) > log(0.001)
 <==> -n*log(2) > -3
 <==> n < 3/log(2) ~=~ 9.96578

f(n) = 3 + (-1/2)^n < 2.999
 <==> (-1/2)^n < -0.001
 <==> n is odd and (-1/2)*(1/4)^((n-1)/2) < -0.001
 <==> (1/4)^((n-1)/2) > 0.002
 <==> -(n-1)*log(2) > -3 + log(2)
 <==> n < 3/log(2) ~=~ 9.96578

Thus points are excluded if and only if n < 10.  Thus there are 
exactly nine points of the graph of f not in your strip.

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