@mehdi jafari evalr is surely the correct thing to use. Indeed, the vote up is mine. If you use lprint to look at the internal structure of the objects generated by the Tolerances package, you will see that they are all of the of the form INTERVAL(........), which are precisely the things that evalr is for. I just find it amazing that there is no example of this on the page ?Tolerances and no mention of the use of INTERVAL.

For example, using the code in the original Question, take a look at lprint(Phi). Then look at lprint(evalr(Phi)).

Does anyone here know if this need to use evalr is documented anywhere? I don't see it mentioned at ?Tolerances . That page merely has a link to ?evalr at the bottom.

@mounir Matrix is spelled with a capital M. You used a lowercase m.

Also, there's no point in including the command M:= Matrix(convert(A, list)) inside the loop. It's just wasted operations. Put it after the loop.

@love maths I followed the link, and I see that Afn means Average(F, f||n), where Average is the procedure with the corrections made by Preben Alsholm. It seems that f||infinity is a Matrix of all ones for the example f0 given, although it is not immediately obvious to me how to prove that. Given that f||n converges to a Matrix with all entries equal, it is obvious to me that "triangle"(f||n) converges to the 0 Matrix.

Your code works for me in Maple 17; s contains all four constants of integration. What does your s contain?

@mounir 

M:= Matrix(convert(m, list));

@sarra The file link is working now.  I see what you mean. The functions appear different for the 5th, 9th, 10th, and 11th values of t. I think that the 5th is normal acceptable variation. But the numerical solution seems to become unstable starting at the 9th value of t. By the 11th value, it appears to be chaotic. I don't know why this happens, and I don't know much about finite-difference methods. For what it's worth, I set the numeric solution to be computed to 100 Digits and it did not change the situation. I know that certain finite-difference methods are inherently unstable under certain conditions related to the relative values of the spatial spacing h and the time spacing k; I don't remember the exact formula. What is the precise name of the finite-difference method that you are using?

@sarra Please try uploading the file again.

The last sum that you give is just the very well known sum of a geometric series:

sum(1/n^x, x= 1..infinity);

Then the nested sum is just a slight variation of the well known

sum((-1)^n/(n-1), n= 2..infinity);

@mehdi jafari You are using indexed variables (a[0], a[1], etc.) in your Answer, whereas the OP is using concatenated variables (a0, a1, etc.) in the Question. While you may argue that indexed variables are better, the distinction should at least be noted.

You haven't defined A anywhere in the problem statement. The only thing that makes sense to me is that A is an N x N matrix. But the stated convergence property would only be true for certain A, those with spectral radius less than 1.

@hillyzz Copying the code is fine, but it is incomplete. I need the values of k_min, k_max, n0, n1, and n2. Also, what is the procedure multiply? Do you just mean to multiply the Matrices? That can be done as P1p.F1p.P2p.F2p.

@DJJerome1976 For some values of theta, filled is interpretted as the area between the curve and the x axis. It ignores the fact that it is polar coordinates.

@Preben Alsholm It seems that the option filled= true doesn't do anything when used with animatecurve.

Please do not edit a Question, making substantial changes, after the Question has been Answered! If you do, then the Answers don't make sense.

The original question was to find the Fourier series for x^2 on -Pi..Pi and to evaluate this at x= Pi to prove that Sum(1/n^2, n= 1..infinity) = Pi^2/6.