First, let me describe a simpler problem I can already prove:

Consider an infinite sequence whose elements are {c(1), c(2), c(3),
....} such that
c(i+1) = a c(i)
for all i=1,2,...,infinity and some fixed real number "a". Consider
the infinite series
S = sum(i=1,infinity) { c(i) / R^i },
where R is some fixed positive number greater than unity. "R^i" means
"R to the power i."

Then Proposition #1 is:
For any pair {a,R} such that the series S converges to a finite real
number, then
S = B c(1)
for some *POSITIVE* real number B. (In other words, convergence of S
implies B >= 0.)
Proof: The proof of this is easy (just consider the geometric
series).

So far, so good. what I want to do is to generalize Proposition #1 to
the following:

Problem: Consider again an infinite series {c(1), c(2), c(3), ...} but
this time suppose
c(i+2) = x c(i+1) + y c(i)
for all i=1,2,...,infinity and some fixed real numbers {x,y}. Consider
again the infinite series
S = sum(i=1,infinity) { c(i) / R^i },
where R is some fixed positive number greater than unity. Show that:
For any triplet {x,y,R} such that S converges to a finite real
number, then
S = E c(2) + F c(1)
for some positive real number E, where {E,F} are both real numbers. In
other words,
show that convergence of S means that E must be non-negative.

[Note: I suspect B must be positive for any {x,y,R} that lets S
converge. However, if this is not generally
true, then my question would be: What are sufficient and necessary
conditions on {x,y} that implies
B must be positive whenever S converges.]

Thanks in advance!