@Alfred_F  You should be aware that the "Maple solution" is just typesetting. The only computation made by Maple is Zeta(2)^2/Zeta(4) = 5/2.  The pictures at the end are screenshots generated by AI, probably Microsoft Copilot.

@Alfred_F The standard method is to use the Moebius inversion. More generally, summing 1/(a^p*b^p), over (a,b)=1, the result is Zeta(p)^2 / Zeta(2*p).

@mmcdara Vote up of course!
I have checked some Maple versions >= 2018 I have installed, but they all do not work!
It would be nice to know when and what has changed (in sum, limit) in the interval 2015--2018.

@Alfred_F For such unusual problems you must guide Maple as I did.
BTW, Mathematica is also unable to compute the limit directly.

@nm  Have you tried a fresh session, and just the simplify commant (no interface(...), restart etc)?

@acer How did you know that

sum( (sum(1/k,k=1..n-1) + 1/n) /n/(n+1), n=1..infinity );

will work? I wouldn't have suspected it!

@nm I meant systematic algorithms. Of course, some empirical algorithms do exist.
Have you tried my example with mma?

@janhardo  A^2 = B^2  does NOT imply A = B.

@sursumCorda PDETools:-Solve works (a bit slower)  without subs too:
PDETools:-Solve(eq,  trial_solution_constants, 'independentof' = t);

@mmcdara Carl has already answered:  (z^(1/n))^n = z by definition of z^(1/n)  for n a nonzero integer.

The problem was caused when I tested "Maple for Screen Readers" (for using the old GUI). At the end, I turned off the "Screen Readers" mode, but for some reason, in the Gui configuration file the entry  "AccessibleMode=true" was left unchanged. After setting manually to "false", the display is normal again.

@dharr Thank you. So, I am not alone.

@dharr Windows 10. So, have you tried to export as .mpl or .tex ?

@acer As this site shows, there are many Maple users interested in typesetting. 
In this case I think It would be a good idea for Maple to include support for LaTeX (see e.g. Geogebra).

@janhardo Why don't you post your code?