How about checking this with ODESteps in Student package ?

@vv 

Thanks, I think you are absolutely right to study the methods of evidence.
The difficulty for me is still how to start translating this into Maple, and your elaboration is an example of how it can be done.

Don't make any progress yet , let me ask Maple AI to come up with the code variants to solve the Basel problem ( The Zeta (2) value ). 
Note: get the impression that the standard AI 3.5 works better for Mathematica ? (not the plugin yet ) 
May try this for the Basel problem for Maple as well...

@janhardo 

It is about comparing series expansion and product expansion of Sin[x]/x. The coefficient of x^2 in the series must coincide with coefficient of x^2 in the product (after expanding the product and collecting powers of x).
Seems to me one of the proofmethods...

simply this equation?
 

eq1 := Product(1 - x^2/(n^2*Pi^2), n = 1 .. infinity) = Sum((-1)^n*x^(2*n)/(2*n+1)!, n = 0 .. infinity) = sin(x)/x;

@sand15 
 

"More generally, ChatGPT does not shine for its clarity (Point 6 is quite obscure)."

You can challenge chatGPT to come up with a better answer and also zoom in on a detail 
I'm using the public chatGPT 3.5 (free) and not the one in Maple itself and therein is a better version chatGPT 4.5?, but don't find it working conveniently in Maple yet 
Also mathematica has developed its own language model plugin and that requires chatgtp 4.5 with a subscription
I wonder how this works in Mathematica 
Ideal would be if Maple would use its own language model focused on Maple content and then you can ask anything, at least it seems to me that this is possible ?

@Axel Vogt 
Thanks, those are great sources of information on the Basel problem.
Don't know which double integral of Apostle that is ?
Via the Student package in Maple with a step solution for 1 integral ( surely a double integral can be solved with repeated integration ? )

@mmcdara 
Thanks, there are not that many faculties numbers to investigate for a given number whether it is a faculties number 

Oh now I see you replied that in your opinion, no faculties numbers can be written symbolically in the series
Well, yes , I do recognise some of them now, so for a proof step well follow then? 

@sand15 
Thanks!, you are quite right that chatGPT is not a really solid foundation of knowledge, but I still find it very helpful to figure out some.
From orginele Euler pdf is quite a chore to get exactly clear why the sum: 1/n^2 (to infinity) = 1/6 Pi^2
Still try in Maple with the wiki examples

Let's look at sin(x) 
Now how to get n! numbers in the serie?

T:=(x,n)->subs(t=x,convert(taylor(sin(t),t=0,n+1),polynom));
   T := proc (x, n) options operator, arrow; subs(t = x, 
      convert(taylor(sin(t), t = 0, n+1), polynom)) end proc


sin(x)=T(x,10)+`...`;
                  1  3    1   5    1    7     1     9      
     sin(x) = x - - x  + --- x  - ---- x  + ------ x  + ...
                  6      120      5040      362880         

@vv 
Thanks, 
"but note that Maple cannot use ithprime symbolically."

I don't think it's a problem, because from comparing 2 finite sequences, Euler could derive something  
So first get the two series according to me.
From the euler product formula you have given, a partial sum can be made 

"1. primes (wrongly suposed to be the set of prime numbers in Maple) is infinite, so it cannot work for Product. Maple rejects even the syntax.
This is euler product formule ....
From this defintion of the zeta function as an euler -produkt formula, this maple code cannot make out the series... should be possible I suppose
Possibly the hypergeometric function is related to the eulerprodukt formula ?

Interesting all those examples on Wikipedia with proof steps.
The question is whether this can also be done in Maple, without too much complicated maths ?
Anyway, the AI generated some maple code and to what extent this can be used ?

@vv 

Thanks, I see in the Maple examples that a closed form for a sequence can also be found and then the convergence of it can be determined
yes, indeed only convergence can be determined from certain sums, apparently. 
If it becomes too difficult to represent an outcome in steps.
If convergence is established, then you can try to solve it further.
The code is generated by AI and I can look at multiple variants in Maple and see step by step where it is "going wrong" and try to find a solution in Maple?

I can try to go further with this code and try to solve the steps 
Note that  coeff is a protected name in Maple 

A plan for solving

@Preben Alsholm 
 

Thanks, yes maybe if it's just about the integral curve itself 
Seeing the line element field I also find illustrative.
Maybe DEplot cannot provide animation ?
There is a separate dfieldplot command and when using DEplot it does provide a line element field.
Understand that with numerically solving an ode, there is always a solution to be found ?
Are nice examples to keep as procedures.