@J F Ogilvie 

For many users the xxx__yyy typesetting convention seems to be just enough. I don't see any reason to sacrifice the table functionality for the sake of typesetting. And even without xxx__yyy: why should it be so difficult to use names with more than one letter; must they be exactly the same as in a published article? This is anyway impossible because in many cases in articles the symbols are context-sensitive.

@tomleslie 

But

simplify(subs(z = -z, sin(z)));  # ==>  - sin(z)

(automatic simplification)

It is only (4.)  which intrigues me because simplify usually distributes over operands.

map(simplify, (sqrt(1-I)*sqrt(1+I)-sqrt(2))*(t+1));  # ==> 0

Nice remark. An even shorter procedure is

MaxL:= (a,b,c,r)-> sqrt(a^2+b^2+c^2)-2*r*sqrt((a^2+b^2+c^2)/min(a,b,c)^2-1);

What about 34234-2*17117 ?

BTW, F(a) = Kitonium's f(a),  but you think f is better.

@Kitonum 

This one is simpler and works for symbolic entries too:

F := a -> piecewise(is(a,odd),{(-a+1)/2,(-a-1)/2},{-round(a/2)});

Do you use a random generator for functions of two variables?

@guras 

@Jjjones98 

Then write S(n) as

Sum((1/k^0.1)*(sin(1/k)-1/k), k=1..n) +  Sum(1/k^(11/10), k=1..n);

The fist sum has a rapid convergence; for the second one use:

asympt(Sum(1/k^(11/10), k=1..n),n);
 

@Kitonum 

Actually the symbolic solution is probably useless (being too long) even without parametric and allsolutions.

You may be also interested in the related Apollonian circles
https://www.mapleprimes.com/questions/222071-Drawing-An-Apollonian-Gasket

Apollonius (of Perga) lived more than five centuries before Pappus.

Do you think that Maplesoft should invest a so much effort for typesetting?
Typesetting is a nice feature but I think that an excessive fine-tuning is too expensive.
I'd prefer these efforts be directed towards maths.
And anyway, Maple is not going to compete with LaTeX.
By the way, wouldn't it be more natural a LaTeX convention in the case of atomic variables?
It is not comfortable to use names like  `#msub(mi("x"),mn("1"))`  (in 1D notation)
(of course a macro() can be used here but even so...).

I also did not noticed the square. But Kitonum posted already a nice and complete solution.

P.S. It will be better in the future to post code instead of pictures.

@sajad 

@_Maxim_ 

I think you forgot index=1 in RootOf. Then it works as expected.
(It's only evalf which takes index=1 by default.)