restart;
Expr:=n->`*`(seq(seq(alpha[j]-alpha[i], j=i+1..n), i=0..n-1));

Example of use:
Expr(3);
                

 tomleslie's method does not work correctly for  t<0 . Below is a fixed and improved version. We use discont option to remove vertical lines where the function is discontinuous. These places can be identified through a dotted line (second plot). The values of the function at points multiple of  2*Pi  can be shown with small solid circles (third plot) :

Download periodic_function.mw

First, we find at what height ( z-value) these surfaces intersect, and then plot them using parametric equations and polar coordinates on the plane:

# z=sqrt(x^2+y^2)
solve({12-z^2=z,z>0});
A:=plot3d([r*cos(t),r*sin(t),12-r^2],r=0..3,t=0..2*Pi,color=khaki):
B:=plot3d([r*cos(t),r*sin(t),r],r=0..8,t=0..2*Pi,transparency=0.5):
plots:-display(A,B, scaling=constrained, axes=normal, labels=[x,y,``],labelfont=[times,14]);

b := 3*(4*x+1)/(diff(f(y),y)*(3*x+1)):
A:=(4*x + 1)/diff(f(y),y) = a:
subs(diff(f(y),y)=solve(A,diff(f(y),y)), b);

Execute

showstat(LinearAlgebra:-IsSimilar);

Example:

restart;
N:=3:
z:=x+I*y:
evalc(z^N);
                                 x^3-3*x*y^2+I*(3*x^2*y-y^3)

Examples.

Code in 1d math:
A:=<1,2; -1,5>;
LinearAlgebra:-Determinant(A);

Code in 2d math:
A:=<1,2; -1,5>;
|A|;

You probably have 2d math by default. Therefore, use the second example.

Edit. If you are not allowed to use a Maple procedure, here is a user-defined recursive procedure that evaluates the determinant of a matrix:

MyDet:=proc(A::Matrix)
local m,n;
m,n:=op(1,A);
if m<>n then error "The Matrix should be square" fi;
if m=1 then return A[1,1] else
add(A[1,k]*(-1)^(1+k)*MyDet(A[2..m,[seq(`if`(j<>k,j,NULL), j=1..n)]]),k=1..n) fi;
end proc:

Example of use:
A:=<1,2,3;4,5,6;7,8,10>;
MyDet(A);
 

You can calculate this integral numerically using a simple procedure as follows:

Int1 := (b,k,r,R)->evalf(Int(exp(-z*(R^2*k^2 - b^2*z)/(R*b))/(z*HeunB(0, k^2*R^2/(b*sqrt(R*b)), R^3*k^4/(4*b^3), 0, -sqrt(R*b)*z/R)^2), z = R .. r));

Example of use:

Int1(0.5, 1, 2, 0.9);
                                                1.462594758

Addition. Below I made an animation of this travel (continuation of the above code). The first frame shows the initially given points  A  and  C  and a green vector that determines the direction of movement:

Back:=display(textplot3d([cA[],"A"], font=[times,18], align=right),textplot3d([cC[],"C"], font=[times,18], align=right),arrow([1,2,4],v,color=green),pointplot3d([cA,cC],symbol=solidsphere,color=red,symbolsize=17)):
Anim1:=animate((display@plottools:-line),[cA,cA+~t*~(cB-cA), color=red,thickness=3],t=0..1,background=Back,paraminfo=false):
Anim2:=animate(display,[textplot3d([cB[],"B"], font=[times,18], align=right)],a=0..1,frames=5,background=display(Back,pointplot3d(cB,symbol=solidsphere,color=red,symbolsize=17)),paraminfo=false):
Anim3:=animate((display@plottools:-line),[cB,cB+~t*~(cC-cB), color=red,thickness=3],t=0..1,background=display(Back,pointplot3d(cB,symbol=solidsphere,color=red,symbolsize=17)),paraminfo=false):
display([Anim1, display(op([1,-1,1],Anim1),Anim2), display(op([1,-1,1],Anim1),op([1,-1,1],Anim2),Anim3)], insequence, axes=normal, labels=[x,y,z], view=[0..14,0..14,0..15], orientation=[-45,75]);

                      

Edit.

Download travel.mw

Below is an example of solving your problem for a set of 10 random vectors:

Download GramMat_new1.mw

by the series command.

Example:

N:=10:
series(cos(x)^n, x=0, N);

        1-(1/2)*n*x^2+(-(1/12)*n+(1/8)*n^2)*x^4+(-(1/45)*n+(1/24)*n^2-  (1/48)*n^3)*x^6+(-17/2520*n+7/480*(n^2)-(1/96)*n^3+(1/384)*n^4)*x^8+O(x^10)

on a simple example from a help:

restart;
with(plots): 
A := Matrix([[2, 1, 0, 0, 3], [0, 2, 1, 0, 0], [0, 0, 2, 1, 0], [0, 0, 0, 2, 1], [0, 0, 0, 0, 2]]);
S := [([1, 2, 3, 4, 5]=~StringTools:-Char~(96+~[$1..5]))[]];
sparsematrixplot(A, matrixview, tickmarks = [S, map(t->-lhs(t) = rhs(t) , S)]);

As far as I know, Maple does not implement a step-by-step solution of differential equations. However, you can find out the type of your equation, for example:

DEtools:-odeadvisor(diff(y(x),x)=(x^2+1)*sin(y(x)), y(x));

 Output:                          [_separable]

For details see help on  ?dsolve,details

Try the following, where  Expr  is your expression to be simplified. If this does not help, then upload your worksheet here.

map(simplify, Expr);

The  map  command forces the application of the  simplify  command separately to each operand of the expression.

Edit.

We see that the sequence  f1(k)  is the sequence of even numbers, starting with 2. Therefore, the sum of the corresponding series is equal to infinity:

H := r->piecewise(r < 1, 1, 0);
f1 := k-> sum(2*(1-H(j)), j = 0 .. k);
seq(f1(k), k=1..10);
expand(sum(2*k, k=1..n));
sum(2*k, k=1..infinity);