November 14th, 2011, 03:37 PM  #1 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Sequence Trajectory [color=#000000]In the figure below we see the first 15 terms of the complex sequence , [attachment=1:23zjs9hc]n15.gif[/attachment:23zjs9hc] where it seems that has an asymptotic behavior. Can you prove it? Plotting for n=100 though, the plot becomes chaotic, something which can be seen in the next figure, [attachment=0:23zjs9hc]n100.gif[/attachment:23zjs9hc] can you figure out what goes wrong? Of course the asymptotic behavior in the first question may vary from computer to computer meaning that the n's for which it holds may be greater than 15.[/color] 
November 14th, 2011, 11:04 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Sequence Trajectory
I don't understand what you're plotting, since it's on a twodimensional graph (rather than a 3dimensional graph for N > C or R > C or a 4dimensional graph for C > C). Both the real part and the imaginary part of the function you defined oscillate rapidly, at least if you take the usual continuation. There are about e * n! peaks and troughs on the graphs from 1 to n. 
November 15th, 2011, 02:21 AM  #3  
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Sequence Trajectory Quote:
when a complex sequence is plotted on a 2D plane, we wright it in the form . I would like to see another try. [/color]  
November 15th, 2011, 05:55 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Sequence Trajectory
You still haven't described what you actually graphed, since your graph is twodimensional and the space you describe (N > C) is threedimensional. I described what's going on with the function (real and imaginary parts are oscillating) but I don't know how that corresponds to your graph. 
November 15th, 2011, 06:17 AM  #5 
Senior Member Joined: Oct 2011 From: Belgium Posts: 522 Thanks: 0  Re: Sequence Trajectory
I think ZardoZ meant in his previous post without i in the argument of cos and sin. This sequence can be plotted on a 2D graph. 
November 15th, 2011, 09:38 AM  #6 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Sequence Trajectory
Plotted parametrically? OK, in that case the problem is just roundoff error. I needed 200 digits or so to get it right. You can see what's going on if you look at the series expansion of exp at 1  the first n terms are ignored (as they make integral multiples of 2 pi), the next one is the main term, and the following terms are inconsequential. 
November 15th, 2011, 09:53 AM  #7 
Senior Member Joined: Oct 2011 From: Belgium Posts: 522 Thanks: 0  Re: Sequence Trajectory 
November 15th, 2011, 11:16 AM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Sequence Trajectory Quote:
 
November 15th, 2011, 11:24 AM  #9  
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Sequence Trajectory Quote:
Quote:
[/color]  
November 15th, 2011, 02:58 PM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Sequence Trajectory Quote:
though. Quote:
 

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