Math 421  •  Fall 2006

Sequential convergence visualized

Copyright © 2004–2006 by Murray Eisenberg.  All rights reserved.

Initializations

Load David Park's Cardano3 application packages:

Needs["Cardano3`ComplexGraphics`"]
Needs["Cardano3`ComplexRoutines`"]

Turn off some annoying warning messages that might otherwise appear about similarly-named objects.  (If you want to see these messages so as to be sure to avoid harm, evaluate expressions like the ones below but replace Off there by On.)

Off[General::spell]
Off[General::spell1]

An example

What is to be visualized is the sequence 
	\(TraditionalForm\`\((1\/\@n\  + \ i\ \ \(n\  + \ 1\)\/n)\)\_\(n\  = \ 1, \ 2, \ …, \ ∞\)\)
and its limiting behavior.

First remove not only any existing value assigned to z, but also any attributes that symbol might still have (and clear the value for n, just to be sure):

Remove[z]; 
Clear[n];

Here's the sequence.  (Notice the use of the pattern variable n_ in the subscript on the left.)

(Subscript[z, n])_  =  
        (1/(Sqrt[n]))  +  I ((n  +  1)/n);

Limit[(Subscript[z, n]),  n  ->  ∞]

You know the theorem that a sequence ((Subscript[z, n])) converges if and only if the sequences of its real and imaginary parts ((Subscript[x, n])) and ((Subscript[y, n])) both converge, and then \(TraditionalForm\`lim\_\(\(\ \)\(n\  -> \ ∞\)\)\ 
          z\_n\  = \ 
        lim\_\(\(\ \)\(n\  -> \ ∞\)\)\ 
            x\_n\  + \ i\ \(lim\_\(\(\ \)\(n\  -> \ ∞\)\)\ y\_n\)\).  Check that here:

{Limit[Re[(Subscript[z, n])],  n  ->  ∞],  
      Limit[Im[(Subscript[z, n])],  n  ->  ∞]}

Now visualize the start of the sequence (the first 100 entries):

ComplexGraphics[{Blue,  PointSize[0.01],  
          ComplexPoint  /@  
            Table[(Subscript[z, n]),  {n,  1,  
                100}]},  
PlotRange  ->  {{0,  
              2},  {0,  2}},  
        FrameTicks  ->  Automatic];

Exercise.  In the plot you should just have generated, it doesn't look like the entries in the sequence are getting very close to i  =  0  +  1 i yet.  So:
(a) try that graphic again but now with the upper bound on n much bigger; and then
(b) also make the "plot window" fit more tightly around the limiting point i by suitably contracting PlotRange.