@vv 

...... of the theorems of Lebesgue, Levi, Fatou. ... (Lit.: Hewitt/Stromberg, Natanson, Shilov/Gurevich) You are right with the reference to "almost everywhere" in a more general type of convergence. Up to now, however, I have assumed that we are working in "classical" analysis. I will bear this in mind in future.

@vv 

...but it fails at the point (1;1), but it is initially a clever solution. Nevertheless, the limit of the double integral over the entire square can be calculated.

My questions are not about finding fault. Rather, I would like to be able to convince myself of the plausibility of the calculations using example calculations. I certainly won't want to see any source code.

@nm 

Thank you very much. Thanks to the logical working structure of Maple that you have described, the solution is testable and reliable. (It is probably an old occupational hazard that I attach importance to such things ;-). )

@nm 

How does Maple arrive at the correct result in a flash? Does Maple calculate or does it look in a database?

I'm asking because a solution using pen and paper is very tiring. On the other hand, it should be known that the integral to be calculated is related to a famous function that has been the subject of a lot of research and publishing.

@Kitonum @Scot Gould

The power of Maple commands is impressive.
But:
Is there a way to make the intermediate steps on the way to the solution visible and verifiable? What do Maple commands do "behind the scenes"?

@Kitonum 

My solution to this old (1878) problem is much more complicated. It was an unwieldy Diophantine equation to solve. I had hoped that Maple would be used for this. :-(

@Alfred_F 

The calculation result from (3) according to (6) assumes that the formation of the limit and the definite integration can be interchanged. Irrespective of the fact that the calculation using the antiderivative leads to the correct result in an elegant way, a formal addition is necessary. For every natural n, the integrand (2) or (3) in [0; 1] is continuous and bounded. According to Heine's theorem, it is then even uniformly continuous. And according to another theorem, the limit and the integral can then be interchanged. Only then does the limit move into the exponents of the integrand and, as proven, deliver the result.
A wonderful task worth remembering :-).

@vv 

I am still fairly familiar with the theorems of analysis relevant to this interesting task. I was confused by the discrepancy with numerical experiments for large n using Mathcad15 (forget it) and derive. I was interested in the numerically comprehensible convergence behavior. I would also have liked to understand the details of the Maple calculation of the limit. But now everything is clear.

This is not a proof of the convergence of the sequence defined according to (2). A calculation in long number arithmetic (limit replaced by large n, derive) yields the number 2.4567 for n=10^12...
Unfortunately, Maple hides the calculation steps that lead to the result 0.0004937... Please explain ;-)

@Kitonum 

Your reference to literature was a pleasure. It also reminded me of my remaining linguistic knowledge, when books by Michlin, Pontrjagin and Shilov/Gurevich were in great demand in the original, for example.

As I am more interested in solution paths and ways of thinking in mathematics than the solution itself, I have tried to create a solution path that is clear and easy to understand for students. It should work with the help of typical school "tools". That is why it looks complicated and unprofessional at first glance - so please don't laugh. As I still make mistakes with Maple, I had to use good old derive and create the attached files from it.

Seite_1_und_3.pdf

Seite_2_und_4.pdf

@dharr ... I thank you for Your solution. It helped me to understand the geometry package a little. This means I am now able to solve my own special private puzzles. In the future I will offer much more difficult problems from my old collection here. These will be suitable for Maple and come from the areas of analysis, geometry and number theory. I hope you like them.

@dharr 

I hadn't expected such a difficult solution. Nevertheless, it helped me to understand the Maple command sequence and to recognize the logic behind the package selection.
The following solution is simpler:
The n-gon has a side length of 1. The interior angle between adjacent polygon sides is known to be (n-2)*pi/n. This angle is opposite the side length we are looking for in the triangle. The side lengths on the sides of this angle (n-2)*pi/n are 1/2. The cosine theorem then gives us the side length we are looking for of the inscribed polygon as 1/2-1/2*cos((n-2)*pi/n) and the limit n--->oo is obvious.

@vv 

Please excuse me. I will try to improve. Thanks for the reference.

@Rouben Rostamian  

...I solved this problem in exactly the same way in 1977 using pen and paper.

@vv 

I would have liked to have had these options 60 years ago. I would have been spared many full wastepaper baskets. ;-)

At the moment, it is very important to me that theoretical backgrounds are also understood in principle. That is why I like to refer to literature in the usual way.

There used to be a humorous saying:
If Kamke doesn't say anything, it can't be solved.