OP wrote "Never understood the proc() without parameters"

In fact, everything is clear here. Parameterless procedures are often used when the number of actual parameters (called arguments) is not known in advance. In this case they are referred with  args (argument sequence)  and  nargs (number of arguments) :

Your generalized example (finds the Euclidean norm for any dimension):

 norm:=proc()
sqrt(sum(args[i]^2,i=1..nargs)) end:

norm(3,4,5);

In the following example, we find the arithmetic mean of a sequence of numbers for any number of these numbers:

restart;
M:=proc()
`+`(args)/nargs;
end proc:

# Example:

M(2,3,5,6);

                                        4

Example:

convert(1.0*a+b, rational);

                           a + b

From the help:
"Important: The evalb command does not simplify expressions. It may return false for a relation that is true. In such a case, apply a simplification to the relation before using evalb."

Your example:

restart;
f := x + 1 = (x^2 - 1)/(x - 1):
evalb(simplify(f));

                                     true

See help for details.
 

This is not a surface but a curve as the intersection of two surfaces (the cylinder  (y-1)^2+z^2=1  and the plane  x=1 ):

Download plot.mw

I don't see any point in solving such a simple problem with Maple. We get the answer immediately only by looking at the drawing without any calculations. SK=KB, KO=1/3*KB so  cos(SKB) = 1/3 .

The code for the pic.

restart;
local O;
with(plottools): with(plots):
A:=[0,0,0]: B:=[1/2,sqrt(3)/2,0]: C:=[1,0,0]: S:=[1/2,sqrt(3)/6,sqrt(2/3)]: K:=[1/2,0,0]: O:=[1/2,sqrt(3)/6,0]:
AB:=line(A,B): BC:=line(C,B): AC:=line(A,C): AS:=line(A,S): BS:=line(B,S): CS:=line(C,S): KS:=line(K,S): SO:=line(S,O): KB:=line(K,B):
T:=textplot3d([[A[],"A"], [B[],"B"], [C[],"C"], [S[],"S"], [K[],"K"], [O[],"O"]], font=[times,bold,18], align={left,above}):  
display(AB,BC,AC,AS,BS,CS,KS,SO,KB,T, color=black, axes=none, orientation=[-50,75], scaling=constrained);

restart;
M:=2*p*q*r*s*t/(u*v*w); 
simplify(M, {p*q/(u*v)=K});

We can of course use  map , but the code will be a little more cumbersome:

A:=[1,2,3]:
B:=[7,8,9]:
map(f@op, [seq([A[i],B[i]],i=1..nops(A))]);

                        [f(1, 7), f(2, 8), f(3, 9)]

restart;
sum((-1)^n/n*x^n, n=1..infinity) assuming x>-1, x<=1;

                                         -ln(1+x)

Edit. You can use the  parametric  option as well:

restart;
sum((-1)^n/n*x^n, n=1..infinity, parametric);

We can use splines for smoothing. You didn’t provide a list of 11 points for your curve, so I restored it by eye.

Download FineCurve.mw

For some reason  assuming  does not always work. Use  assume=even  instead:

restart;
expr := Sum((-1)^n - 1, n = 1 .. infinity):
simplify(op(1,expr), assume=even);

                                        0

Edit. Probably something related to the levels of calculations here. For example, the following option works:

restart;
expr := Sum((-1)^n - 1, n = 1 .. infinity):
op(1,expr);
simplify(%) assuming n::even;

What you want to do is easy using the commands  plottools:-rectangle , plots:-inequal  and  plots:-textplot :

Download Venn.mw

Use  style=line  option for the surfaces:

plots:-display(plot3d(2*(1-x^2-y^2), x=-2..2, y=-3..3, style=line, view=0..2, grid=[20,30]), plot3d(0, x=-2..2, y=-3..3, style=line, grid=[20,30]), plots:-spacecurve([cos(t),sin(t)+1/2,0],t=0..2*Pi, color=black, thickness=2), scaling=constrained, axes=none);

Use additional graphic options that can significantly improve the quality of plotting. The plot size can be changed with  size  option, and the style (including size) of the legends is controlled by  legendstyle  option. As an example, look at the plotting 4 basic trigonometric functions on one plot by default and using additional options:

Additionally, I replaced the vertical lines that Maple draws by default at break points with vertical dashed black lines and indicated them in the legends. The colors of the graphs have also been changed.

Download options.mw

I did not find a command in Maple for expanding a function into a Fourier series. But here's a simple procedure that does it for a function on the segment  -Pi .. Pi . I leave it to you as a simple exercise to modify it for a function given on an arbitrary segment.
 

Edit.

Download Fourier_series1.mw

Here's another easy way. Thу  ZigZag  procedure arranges the matrix indices in zigzag order. The matrix can be arbitrary (not necessarily square).

Download ZigZag.mw