Does the sum of the reciprocals of composites that are $ \le $ 1 The sum itself:
$$ \frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+ \frac{1}{15}+ \frac{1}{39}... \le 1 $$
These are all sums of reciprocals of composites that can be closest to 1. Notice that the term after $\frac{1}{15}$ is $\frac{1}{39}$ rather than $\frac{1}{16}$, or any other reciprocal of a composite number between 15 and 39 for the reason that it would make the sum up to that point $\gt$ 1. That is a description of this sum from the most basic level. 
Sum Parameters:
-If 1 is reached at an exact given point, the sum stops and thus making the sum finite.
-The sum must be $\le $ 1
-All terms must be the reciprocal of composite numbers.
-To find the next term in the list, you must find the largest reciprocal (that fits standards) that makes the sum $\le$ 1, and NOT the term that best fits the sum. For example, if you were to need to add $\frac{1}{x}$ and $\frac{1}{y}$ to make the sum exactly 1, but $\frac{1}{a}$ was the largest and still fit the rest of the parameters, then you would have to stick with with $\frac{1}{a}$. (Even if $\frac{1}{a}$ was a really ugly term to add at that point)
Questions:
1) With these rules in place will the sum be a finite or infinite sum? If this is an infinite sum than the terms will always get closer and closer to 1, but never actually get there with a finite number of terms. With an finite sum, then at a quantifiable number of terms does the sum reach 1.
2)If the sum turns out to be infinite, than please tell me why.
3)If the sum turns out to be finite, list the terms. If the sum is ridiculously large, then prove that it is finite.
Thank you

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. 
