Use assumptions or formal sums

sum(x^n,n=0..infinity) assuming abs(x)<1;
sum(x^n,n=0..infinity,  formal);

Download int.mw

It is not possible to use plots:-animate because HeatMap is implemented to display a background picture.
Explore can be used, but I am not sure whether it works in Maple 2016. In Maple 2017+ it's ok.

restart;
p := k -> Matrix(3, 3, (i,j) -> (i+j+k) mod 3):
f:=proc(k) Threads:-Sleep(0.2): Statistics:-HeatMap(p(k)) end:
Explore(  f(k), k=0..8, animate,loop, autorun);

When a variable is implicitely declared local, a warning message appears.
The rules for this are simple, see ?local

In

foo:= proc() 
  local x;
  plot(sin(x),x=-Pi..Pi); 
end proc:

local is indeed needed, otherwise x:=10  at top level will produce an error.

Alternatively you may use
plot(sin('x'),'x'=-Pi..Pi);

All the floating point computations are done with 1 significant digit. So, sqrt(5) is approximated to 2.  and  (2. -1.)/2.  is 0.5.

See also:

restart;
Digits:=1;
                          Digits := 1
a:=[seq(i/10., i=1..9)];
       a := [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]
b:=[seq(i/10., i=9..1,-1)];
       b := [0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1]
add(a) = add(b);
                            6. = 4.
evalf[2](add(a)) = evalf[2](add(b));
                           4.5 = 4.5

Maple (at least the recent versions) can compute directly the double integral (without passing to polar coordinates):

int(sqrt(x^2+y^2), x = 0 .. B/2, y = 0 .. b/2) assuming b>0, B>0;

S:=simplify(eval(SA/t, [x=y*t, Log[t]=1/L])):
collect(S,y, expand);

Now you have a polynomial in the variables (L,y) with rational coefficients and you want to guess its coefficients or a generating function. Why do you think that a simple generating function does exist? A more realistic approach would be to come back to the original problem (which produced the expression) and try there.

.

n >= ceil(fsolve(1/(n+1)! = 0.00001));
                             8 <= n
n >= ceil(fsolve(1/(n+1)! * exp(0.1) = 0.00001));
                             8 <= n

Seems to be a bug. Workaround:

ex:=D[2](eta)(t,x)+D[2](phi)(t,x,0)+D[1](phi)(t,x,0);

inds:=[indets(ex)[]]:  cinds:=map(convert, inds, Diff):
subs(inds=~cinds, ex);

Download Interpolate.mw

a:=sqrt(6)/6 + sin(5/6) + exp(6/7) + 567:

f := u ->  `/`(subs(6=b, [op(u)])[]):
subsindets(subs(6=b,a), fraction, f);

For non-numeric coefficients you may use
signum(lcoeff(expr));

@AliahNiu 

For such functions Maple cannot help much, we must use some maths.
You have two (possible) singularities at x=0 and x=Pi.
For the integral to exist near 0 a necessary and sufficient condition is  (a+1/2)*p+1-d < 0.
Near Pi the condition is (b+1/2)*p+1-d < 0.
So, the integral exists iff both conditions hold.

Note. Here the integral was considered a an improper Riemann one (in this case this being equivalent with Lebesgue integrability). If you want standard Riemann integrability (equivalent here with continuity) the conditions are similar but more restrictive, namely <= -1  instead of  < 0.

Download check-pde.mw

Edit. Actually for u() given by Maple it is easy to compute  u(1, 1/2)  = 1/12, so definitively wrong!

I'll choose the space for you.

u1 := (x, y)-> 1-exp(a*x)*cos(2*Pi*y):
u2 := (x, y)-> a*exp(a*x)*cos(2*Pi*y):
Norm2 := u -> int(u^2, 0..1,0..1)^(1/2):
simplify( Norm2(u2)/Norm2(u1) );