@Hermitage What tools would you have available on a test? Perhaps a graphing calculator? Maple? Although the problem is somewhat tedious to do analytically, the most basic plot shows that the range is [0, +infinity).

@Carl Love Being able to upload worksheets for display in a post is of fundamental importance for MaplePrimes, IMO. Are there any progress reports on returning this functionality? Are you planning to return this functionality?

@adel-00 Remove the line tau:= Pi from the top of the program. That's all that you need to do. I said that before.

The table in your question is not a Cayley table. There are no repeated elements in any row or column of a Cayley table.

@candy898 The natural logarithm of x is written ln(x), not In(x): letter L for Logarithm.

So, did you mean for (x-1) to be in the denominator?

@candy898 What does In(x) mean? There is no standard function with that name. Also, it is not clear if you mean for (x-1) to be in the numerator or denominator. If you don't know how to make the symbols, then type it out in English words.

Yeah, I know what an integral is, and I know that you mean that.

tau:= Pi:
P1:= plot(Spec, -10..10, axes=boxed, title=tit, color=black, font=[2,3,18],
            thickness=2, tickmarks=[3,3], titlefont=[SYMBOL,14], font=[1,1,18],
            linestyle=1
       ):

Normalize:= proc(P::specfunc(anything, PLOT))
local A,Smax1;
     A:= op([1,1], P);
     Smax1:= max(A[..,2]);
     if A::list then  A:= Matrix(A)  end if;
     A[..,2]:= A[..,2]/Smax1;
     subsop([1,1]= A, P)
end proc:

P1:= Normalize(P1):
for k from 2 to 5 do
     tau:= k*Pi;
     P||k:= plot(Spec, -10..10);
     P||k:= plottools:-translate(Normalize(P||k), 0, k-1)
od:
display([P||(1..5)]);

@JohnS What you want to do is essentially step through the variables in the basis of the linear solution, setting each to 1 and the rest to zero. I've written some code to do this; it's just a few lines. If you provide a complete example of the equations to solve and the Z1, ..., Zn equations, then I'll post my code.

This would be so much easier if numerical summation had a digits option akin to the option for numerical integration. The reason that the sum doesn't evaluate numerically for all x is that for some x it can't get Digits accuracy. It would be easy for digits < < Digits.

The numeric summation of this series can be accelerated using a technique based on the integral test, as the summand is eventually decreasing and has a symbolic antiderivative. I've written a procedure to do this numeric summation and it has an equivalent of the digits option. But it is much slower than Kitonum's solution, so I'll save it for later. However, it will numerically sum some series that evalf(Sum(...)) won't touch for any x.

@rab285 Is your Matrix 383 rows by 3 columns?

Please post your code, showing an example of the phenomenon that you describe.

@Deltafee You need a semicolon after Array(1..2).

@adel-00 It should be 4*Pi and 5*Pi, not 4Pi and 5Pi.

Do you want to plot two curves on the same axes? or on different axes side-by-side? Using an Array will make them side-by-side.

You may just be missing multiplication signs in your plot commands.

@adel-00 The 2 stands for the 2nd column of the data matrix, which is the y-coordinates. You do not have to change anything inside the procedure.

I can't tell you why it doesn't work for your P4 and P5 because you haven't said how you generated them.