simpSum:=proc(S::specfunc(Sum))
local o1:=op(1,S), eq:=op(2,S), n,a,b,c;
if not type(eq, name=range) then return S fi;
n:=lhs(eq); a,b:=op(rhs(eq));
c:=max(select(type, [solve(o1,n)], integer));
if c>=a and c <= b then
  sum(o1, n=a..c-1) + Sum(o1, n=c+1..b)
else S fi;
end:

simpSum(Sum(n*x^n, n = 0 .. 20));

S:=Sum(n*(n-4)*x^(n^2), n = 0 .. infinity);
simpSum(S);

S := Sum(epsilon^k*v(k,t)/k!, k = 0 .. infinity);

S1 := diff(S, epsilon);

                       
 S1ok:=simpSum(S1);                   

limit(%, epsilon=0);

The result is correct. diff(x^n,x)  is 0 for n=0, not for n=1.
diff(Sum(x^n, n = 0 .. infinity), x) is correct too. Note that diff(Sum(f,...), x)  simply applies diff to f.
Of course, it is possible to simplify by eliminating the 0 term, but unfortunately it has to be done by hand (if you want).

S:=diff(Sum(x^n, n = 0 .. infinity),x);
eval(S, 'n = 0 .. infinity' = 'n = 1 .. infinity');

(In this case you may simply use eval(S, 0=1).)

I think that setting interface(typesetting=extended) solves the problem, but I cannot check this in Maple 2015.
But another problem is that the continued fraction appears as 1/( 1/... + 1). To change this, use inert operators g := x -> 1/(1 %+ x)

1 + something   is not  1.  It corresponds to Not(something).

with(GraphTheory):
G := RandomGraphs:-RandomGraph(8, .5):
u:=3;
DrawGraph(G);
add(map2(Degree, G, Neighbors(G,u)));

MROUND := (x,m) -> round(x/m)*m;

(m can be noninteger)

Here is a more general and robust (I think) procedure. I borrowed from mmcdara the examples and the idea to consider only rational functions.

Download c3.mw

(mapleprimes does not render all the output)

restart;
f:=x->1/(x^2-2*x):
P:=plot(f, -4..4, color=red, discont, view=-4..4, scaling=constrained):
#normal at (a, f(a));
y - f(a) + 1/D(f)(a)*(x-a):
# center of the circle is (1,b)
b:=solve(eval(%, x=1), y):
(b-f(1))^2 = (1-a)^2 + (b-f(a))^2:
A:=select(u->is(u>2), {solve(%)}):
for a1 in A do
  b1:= simplify(eval(b, a=a1)); r1:=abs(b1-f(1));
  Q:=plot( [1+r1*cos(t), b1+r1*sin(t), t=0..2*Pi]):
  print( plots:-display(P,Q, title=typeset(Center=[1,b1],`, r`=r1)) );
od:

You can make a "custom lightmodel" using color. 

restart;
g:=exp(-x^2-y^2):
grad:=[diff(g,x),diff(g,y),-1]:
grad:=-grad/~sqrt(add(grad^~2)):
light:=[-1/sqrt(3),-1/sqrt(3),1/sqrt(3)]:
c:=add(grad*~light):
LED := plot3d(<3 + cos(t)*cos(s), cos(t)*sin(s), sin(t)>, s = 0..2*Pi, t =-Pi/2..Pi/2, 
              color = "Yellow", style=surface, lightmodel = none):
plots:-display(LED, plot3d(g, x=-2..2, y=-2..2, color=[1,1,c, colortype=HSV], style=surface),scaling=constrained);

subs does not evaluate. If you use

Cq:= proc(x,y)
  evalb(x=eval(y,{q2=q1,q1=q2}))
end proc;

everything works as expected. 

The integrals do not make much sense.
You integrate wrt RR = 0..R an indefinite integral. But the indefinite integral is determined up to an (additive) unknown constant.
So, the final result will be also up to a constant, hence it is useless.

You have the derivative of a function (defined by a piecewise expression) at the point t=0 (not an equation!).
The function is not derivable at this point because it is not correctly defined for t<0  (ln(0)).
You probably want the right-derivative, but Maple has only two-sided derivatives.
(It can be computed as a right-limit if you want, or simply remove the first branch in piecewise).
 

It actually works. I added the list type and set the base b to 10.

restart
`&*` := proc(A::list, B::list, b := 10, d := 4) 
    local i, k, c := 0; 
    [seq(irem(c + add(A[i]*B[k+1-i], i = 1 .. k), b, 'c'), k = 1 .. d)]
 end proc:
[4,3,2,1] &* [5,4,3,2]
#                          [0, 3, 7, 3]

1234*2345 mod 10^4;   # check
#                              3730