evalf~([solve(z^5-5*z^4-z^2-2)]);
[argument, abs]~(%);

But it is a correct answer:

limit(n^(3/2)*sum(2*k/(2*k+3), k=1..n), n=infinity);

                                    infinity

In Maple  e  is just a symbol, not  2.71828...

Should be:

int(exp(-x)*sin(x^2)/(2+x), x=1..infinity);
evalf(%);

Also, do not use square brackets to group expressions. They serve to form lists.

If you already know the result, you can just do :

is(abs(u+v)^2 + abs(u-v)^2 = 2*abs(u)^2 + 2*abs(v)^2);
                                          true

Your matrix equation can easily be reduced to a linear system:

gc();
restart:
with(LinearAlgebra):
A := <<a__11|a__12>,<a__21|a__22>>;
P := <<p__11|p__12>,<p__12|p__22>>;
Id := <<1|0>,<0|1>>;
eqn := Transpose(A).P+P.A =~ -Id;
solve({seq(seq(eqn[i,j], j=1..2), i=1..2)}, {p__11,p__12,p__12,p__22});
 

Although your system is linear with respect to the unknowns, it has 10 parameters. Even if Maple wrote formal expressions for the roots through the determinants, we would get huge expressions useless for use. I remind you that the matrix determinant 15 by 15 has in general 15! = 1307674368000 terms. Therefore, it is reasonable to solve the system by specifying the values of the parameters. See the example in the file.

System_new.mw

In 2d math mode that you use, the easiest way to calculate the determinant is to use two vertical bars, which you can enter from the keyboard:

restart;
u:=u0(x,y)+z*u1(x,y):
v:=v0(x,y)+z*v1(x,y):
A:=<a,b; c,d>;
B:=<diff(u,x), diff(v,y)>;
C:=A.B;

General approach in applying to your example:

restart; 
F := ((1/2)*a[2]-a[1]+(1/2)*a[0])*n^2+(-(1/2)*a[2]+2*a[1]-(3/2)*a[0])*n+a[0]; 
N := 3; 
var := [seq(a[n-1], n = 1 .. N)]; 
factor~([seq(op(k, foldl(collect, F, var[]))/var[N-k+1], k = 1 .. N)]);

This is a simple arithmetic:

480*1/(5+1+1/4);
                                      384/5

So the answer is $76.8

A:=<0,1,1,1,0,1; 1,0,0,0,1,1; 1,0,0,0,1,1; 1,0,0,0,1,1;0,1,1,1,0,1; 1,1,1,1,1,0>:
evalc(LinearAlgebra:-Eigenvalues(A));  # Symbolic (exact) result
evalf(%);  # Approximate result
                             

L:=[[TC,DB], [], [TD,JK], [IW,CM], [], [KJ,DJ]];
remove(s->s=[], L);  # or
remove(`=`, L, []);

The first equation is set incorrectly. See the corrected file  SimEquations_new.mw

First we inserted the row  R  into the matrix  A, and then inserted  the column  C  into the matrix  A1:

A:=LinearAlgebra:-RandomMatrix(6, generator=0..9);
R:=LinearAlgebra:-RandomVector[row](6, generator=0..9);
C:=LinearAlgebra:-RandomVector(7, generator=0..9);
A1:=<<A[..3], R>, A[4..]>;
A2:=<A1[..,..3] | C | A1 [..,4..]>;

Edit.

To plot space curves, I usually use  plots:-spacecurve  command, because it gives the best quality, and instead of transparency I select the appropriate  orientation:

restart;
PP:=0.8707945038*exp(-50.00000000*(m-.842e-1)^2+2.745342070*(m-.842e-1)*(a-2.3722)-.1046792095*(a-2.3722)^2):
A:=plots:-spacecurve([a, -0.2, eval(PP,m=-0.2)+0.005], a = -5 .. 8, color=red, thickness=4):
B:=plot3d(PP, a = -5 .. 8, m = -0.7 .. 0.7, style=surface, color=khaki, numpoints=5000):
C:=plot3d([a, -0.2, z], a = -5 .. 8, z = 0 .. 0.9, style=surface, color="LightBlue"):
plots:-display(A,B,C, orientation=[-30,70]);

Edit.