After reviewing your feedback I went back to my worksheet & came to the realization that my R2 is indeed correct.  However, to get R2 to become the sinc squared function the sin term needs to drop out.  That does happen when n is any integer.  I missed this probably due to myopic fatigue.

@Markiyan Hirnyk 

PS. In your attempt try the AllSolutions options in the integrals. Another hint is to assume T> 0  T > 1 in the beginning.

I do not follow your statement here.  Can you elaborate further?

@Rouben Rostamian  

Thanks Rouben, your response provoked me to think about this more extensively.  You basically confirmed my assessment that the n=0 term is 1/4.  So why is that?  Explanation is included in the link below:

asymmetric_triangle_0th_order_Fourier_coeff.mw

I have made some corrections to my earlier worksheet which can be accessed by the link below.  It looks like I accurately assessed the n=0 term = 1/4 based on the plot generated for the Fourier series closely matches the actual function 1-t.  Nonetheless, the analytical solution for the n=0 term suggests that it is UNDEFINED.

asymmetric_triangle_fourier_series.mw

@rlopez 

What packages are speaking of & how can I get access to them?  The function 1-x is pretty basic.  Is there another way to generate the apporopriate Fourier series?

Check the link:

Sq_wave_Fourier_transform.mw

@vv 

So in essence the sinc function is defined on both sides of t=0, but not at t=0?

Thanks to both of you, Tom & Rouben.  I do not think I ever really comprehended the limitations of executing analytical computations digitally.  This issue I stumbled upon is truly valuable.  Both of your explanations were excellent & enlightening.

Once again THANKS!!!

@Rouben Rostamian  

Even if the sig figs is only 10 digits I still do not understand why it would not yield a result = 1.862645149e-08 instead of 0?

I am preparing to do this in MATLAB.  I was hoping I could do it in MAPLE.

Check the link.  This crudely achieves what I am looking for, but it is done in 3 separate MATRIX stages.  I then import the data into MATLAB and merge the 3 matrices into one.  Is there a way to achieve this in a continuous one step command in MAPLE?  This would require the ability to iteratively change the time step and the range for t as a function of the k value.

test.mw

@Carl Love 

"The way that you defined S3, it is an expression, not a procedure. Yet in the Matrix command you invoke S3(k) as if it were a procedure. Consequently, the current Matrix is filled with garbage, as you'll see if you inspect a few random entries."

I uploaded the corrected file with the link below.  It turns out that I copied and pasted various portions of my native .mw document, and I missed a critical one.  That is the cause of "garbage".  I should have tested the output prior to submitting the question.  Something to heed to next time.  So check out the corrected file.  Can I retroactively delete the earlier bad file & resubmit the original question with the corrected file attached?

MATRIX_loop.mw

@acer 

Well thanks for deleting my other posting.  You could have at least copied and pasted it into this posting.  I have now lost what I said & the corrected .mw file!!!!

I now have to spend time to recreate it.

@Carl Love 

"Why do you want the result specifically to be a Matrix? You have the rows indexed by time and the columns indexed by frequency. Using a Matrix forces the number of entries to be (1 + number of times) x (1 + number of frequencies). By using a table instead of a Matrix, you wouldn't be restricted to the rectangular format; you could have separate times for each frequency."

I suppose you might be right, but I have never used the table command.  Maybe you could provide a sample?

"The way that you defined S3, it is an expression, not a procedure. Yet in the Matrix command you invoke S3(k) as if it were a procedure. Consequently, the current Matrix is filled with garbage, as you'll see if you inspect a few random entries."

I am not sure about "garbage" it seems to work on my computer.  All of the elements of the matrix have numerical values.  Maybe something happened to the code when I uploaded it.  I will download the file on this posting & check it.  If something is wrong I guess I will have to resubmit my question with a new heading.

"All you're doing is numerically evaluating S3 and different values of t of k. So why isn't having a procedure S3 of parameters t and k good enough?"

It may be, but I am not familiar with what you are calling a "procedure".  Maybe you can elaborate on that?

I have no idea why someone would give a THUMBS UP on my question without any replies.  Regardless, if my question is NOT CLEAR please let me know if I can elaborate further.  The need is if the frequency parameter k becomes sufficiently high then to resolve that oscillation the time step must become smaller.  For example, an oscillation with a frequency of 1000 Hz a time step of 0.01 is not adequate to monitor the signal.  I believe this is called "aliasing".

So my question is can I change the time step in the MATRIX command based on the k value?  Keep in mind that the time window must decrease accordingly with the time step or the file size will balloon.  I know I can do this in MATLAB, but would like to try this in MAPLE.

THANKS again