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Problem with plotting and simplifying a function

• To: mathgroup at smc.vnet.net
• Subject: [mg100478] Problem with plotting and simplifying a function
• From: new one <mail726 at gmail.com>
• Date: Fri, 5 Jun 2009 03:02:27 -0400 (EDT)

I have a question that troubles me , and I'll appreciate your advices.
Given Sum of : X_N(t) =Sum [ Exp(j2*Pi*k*t) ] with lower limit -N and upper limit +N , -2<= t <= 2
I need to calculate x(t) ,so I used Euler :
X_N(t) = Sum e^(j2*pi*k*t) = Sum [ cos(2pi*k*t) + j*sin (2pi*k*t) ].

Now ,if I'm mistaken the sinus part becom equal to zero , because the cycle
of sinus is PI . But I have a problem with the Cosinus part , how can I
simplify it ? the "t" parameter interrupts with the simplifying ......
maybe separating it to conditions ? N=2R ------> even , N=2R+1 --> odd ?

thanks in advance