@whtitefall 

Use:
l:=1/2;
evalf(DD(1)); 

That's why I have suggested to define R := (x,h) -> ...  etc

 

@one pound 

It is useful for asymptotics; my objection was for using "=" instead of "~".

@Adam Ledger 

f(oo) = oo implies f(k)>0 for k large enough, so your question is answered.
A bound can be computed using x -1 < floor(x) <= x; ==> S subset {1,...,527}. Then use Maple.

In (I) you use divisibility in Q* which is unusual; anyway, the third implication is false.

For (II)  a counterexample is expected.

@tomleslie 

Download distance-geometry.mw

@kuwait1 

It is easy to see that the integrand always has a pole in [0,Pi] for n::posint.

@tomleslie 

That was the purpose of the example. There is no tetrahedron!
(A triangle cannot have the sides 1,1,3).
You should turn both disagree into agree :-)

 

@Carl Love 

I know, but S[n] etc have huge values and for larger n the situation is worse.

Maybe the approximation is valid only near t=0. It's a kind of Picard iteration.

The "homotopy" S_n approch leads to something like:

restart;
N:=5e7;
beta:=0.00001;
mu:=0.02;
sigma:=45.6;
gamma1:=73;

S[0]:=t -> 12500000;
E[0]:=t -> 50000;
J[0]:=t -> 30000;
R[0]:=t -> 37420000;

for n from 0 to 2 do
S[n+1]:= unapply( S[n](t)-int(D(S[n])(w)- mu*N+beta*S[n](w)*J[n](w)+mu*S[n](w), w=0..t), t):
E[n+1]:= unapply( E[n](t)-int(D(E[n])(w) - beta*S[n](w)*J[n](w)+mu*E[n](w)+sigma*E[n](w), w=0..t), t);
J[n+1]:= unapply( J[n](t)-int(D(J[n])(w)- sigma*E[n](w)+mu*J[n](w)+gamma1*J[n](w), w=0..t), t);
R[n+1]:= unapply( R[n](t)-int(D(R[n])(w)-gamma1*J[n](w)+mu*R[n](w), w=0..t), t);
od;

plot(S[3], 0..70);

but does not seem to approximate the solution.

@tomleslie 

1. If you don't add the triangle inequalities, your seq test does not give a correct answer.
You can check this for  test:=[1,1,3,5,1,3];

2. The positive_definite approach is correct only for the Cayley-Menger matrix; it does not work for your  McCrea.
So the correct procedure is

fM:=proc(a,b,c,d,e,f) # a>0 etc
<0,a^2,b^2,c^2,1; a^2,0,f^2,e^2,1; b^2,f^2,0,d^2,1; c^2,e^2,d^2,0,1; 1,1,1,1,0>;
LinearAlgebra:-IsDefinite(%, query = 'positive_definite')
end;

So, it's not a matter of excessive bother.

Best regards,
V.A.

S_0(t)  etc  are needed. Probably you mean  S_0(t) = 12500000 etc.

Why don't you write your equations in Maple to be copy-pasted? Do you prefer to be typed by the forum members?

@digerdiga 

Maple 2016: For infinite sums, Maple is now more careful regarding potentially divergent parametric sums.

@digerdiga 

Mathematically the equality is not true without restrictions. The default option is now formal=false.

@nm 

But e.g. dsolve cannot solve the simple:

dsolve(diff(y(x),x)+Int((y(x)),x=0..x)=x,y(x));

and also

dsolve(diff(y(x),x)+Int((y(x)),x=0..1)=x,y(x));

@acer 

For an ODE, y(x) under Int should be (probably) also rejected.