@bstuan The limit being 2 is fine for proving divergence. See https://math.uchicago.edu/~tghyde/LimitComparison.pdf

@bstuan You wrote:

• My mistake! Sorry for my incorrect word usage.I often use the word "Convergence" to generally refer to the convergence or non-convergence (divergence) of a series or a improper integral.

I think that your original phrasing "convergence property of the following improper integral can be checked by the comparison method" is understandable.

@tomleslie The OP needs to find a function g(x) such that 

• there exists an a > 0 such that 0 <= g(x) <= f(x) for all x in the open interval 0 < x < a,
• g(x) has an easy-to-find antiderivative,
• int(g(x), x= 0..a) = infinity,

in order to prove that int(f(x), x= 0..1) diverges (by comparison test).

Here is a procedure to do a single edge contraction:

Contract:= proc(G::Graph, e::And(set({string, symbol, integer}), 2 &under nops))
description "Remove an edge from G by combining its end vertices.";
uses GT= GraphTheory;
local 
    V:= GT:-Vertices(G), n:= nops(V), J:= table(V=~[$n]), E:= rtable(op(4,G)), 
    u:= J[e[1]], v:= J[e[2]], P:= v+1..n, R:= [1..v-1, P]
;
    E[u]:= (E[u] union E[v]) minus {u,v};
    GT:-Graph(V[R], subsindets(subs(v= u, E[R]), integer[P], `-`, 1))
end proc
:
GT:= GraphTheory:
G:= GT:-SpecialGraphs:-PetersenGraph():
GT:-Edges(G)[1];
                             {1, 2}
G1:= Contract(G, {1,2});
G1 := Graph 10: an undirected unweighted graph with 9 vertices and 14 edge(s)

plots:-display(GT:-DrawGraph~(<G | G1>));

Since we're always going to do two edge contractions, some efficiency can be had by doing them both at once. But I'd rather finish the algorithm first and come back to that later.

@lcz This passage from page 197 of the paper that you referenced explains how to reduce the computation of M to a max-flow problem. This is also what I would've come up with myself.

• We shall show that M(e_i, e_j) for a given pair of independent edges e_i and e_j can be computed by solving a network flow problem.
  We proceed with the computation of M(e_i, e_j). Let e_i = (u, v) and e_j = (x, y). Note that the endvertices u, v, x, and y are all distinct as we are assuming that e_i and e_j are independent edges in G. Now, contract the edge e_i in G by identifying its endvertices u and v to obtain a single vertex called s. Do the same thing for edge e_j and call the corresponding vertex t. The resulting graph will be referred to as G’. We now define a network graph N from G’ by letting vertex s be its source and vertex t be its sink. Then, each edge is replaced by two oppositely directed arcs, the capacity of each arc is set to unity. It is not difficult to see that the value of a max-flow function in network N is equal to M(e_i, e_j) in graph G.

What is the real-valued function M used in the algorithm's pseudo-code? And what is the set-valued function I? I guess that I(v) is either the set of edges connected to v or not connected to v. I have no guess for M.

@rockyicer As you likely realize, what you're currently asking about has nothing to do with the mathematical aspects of the solution. Rather, I did it solely for the display aspect, to put the variables back the way that you originally had them.

Before I give a more-thorough Answer, tell me which (if any) of these Maple commands you already have a basic understanding of: type, indets, subsindets, cat, op, op(0, ...).

In my command above, patindex(anything) is the type of subexpression to be changed. The pat stands for "patterned" (which I think you figured out already). So, it looks for indexed expressions with a certain type pattern in the indices. But, since I specifed anything, it'll match any indexed expression. So, I could've (and for sake of simplicity, I should've) used indexed instead of patindex(anything). See the help pages ?type, ?type,structure (for patindex), ?type,anything, ?indexed, ?type,indexed.

Supposed that n is an indexed name, for example n = x[1,2]. Then op(0, n) = x, op(n) = (1, 2), and op(0.., n) = (x, 1, 2). (There's a distinction between indexed and subscripted. Subscripted refers to the form of typographic display of an expression, but indexed refers to its internally stored structure.) The op stands for "operands". It's the most-fundamental Maple command for deconstructing expressions. See ?op and ?name.

The cat stands for "catenate". It's the Maple command for building symbols and strings. So, using the same n as above, the cat command becomes cat(x, (1, 2)) ==> cat(x, 1, 2) ==> x12. See ?cat.

Regarding the length and complexity of my command: It could be replaced with

S:= simplify(solve(Eqs, Svars));  #Do the algebra.
subsindets[2](S, indexed, 0.., cat@op);  #Do the typography.

The [2] after subsindets causes the indexed subexpressions to be passed as the 2^nd argument to the transformer, cat@op, with 0.. being its (constant) 1^st argument. See ?subsindets.

@nm You're right about the right directional limit, but I have some doubt left about the left: limit(ln(x)/x, x= 0, left) returns infinity. I expected (1 - I)*infinity.

@bstuan Part b is false. The simplest counterexample that I can come up with is 

A = [ 0  1 ]
    [ 0  0 ]

B = [-1  0 ]
    [-1 -1 ]

The complaint that you get from MaplePrimes is advisory, not prohibitive. In the case of Maple Flow, there's no other convenient way to post your work, so just ignore the complaint and post images anyway.

@acer The fact that dsolve is shown to run without error with option arbitraryconstants= subscripted indicates that the OP is using a quite-recent version of Physics updates.

@rockyicer Good job, and thank you. Yes, I had guessed that including equ6 and using your original equ5 would give it enough information to eliminate f0, but I didn't explore that. I'm glad that you did.

A few hints for part a:

1. A square matrix A such that A^k = 0 for some positive integer k is called nilpotent. You should be able to find some useful details about nilpotent matrices by web searching.
2. There's nothing special about 2023 here; any positive integer will do. In other words, if A is nilpotent, then I - A is invertible.
3. Using the distributive property of matrix multiplication, simplify (by hand, not by Maple) 
   (I - A).(I + Sum(A^k, k= 1..2022)). 

@mmcdara Before one proceeds too far along the lines that you suggest (file transfer, etc.), it should be pointed out that Maple has had a built-in Python interpreter since Maple 2018. I know that you use Maple 2015 and may not be aware of this.

See help page ?Python.

@mmcdara Thanks, I voted up for yours also.

I just made a very small modification to my Answer to have Ia1 expressed in terms of Va1 per the OP's request. The modifcation is to subtract Va1 from the Svars.