Does the sum of the reciprocals of composites that are $ \le $ 1
The sum is finite. The next term is the last: $1/6552$.
EDIT: Consider the more general problem with a rational target $T$.
I tried some random choices of target; for the target $105/37$ I was not able to produce a finite sum (after $204$ iterations the denominators were so big that Maple was having trouble with the primality testing). So I'm not convinced that the sum will always be finite.