Collatz Conjecture and Summation

[Unity+]

Well, with my fascination of the Collatz conjecture, I might as well test other things with it.

 

When I combined some concepts to make one. So, here is what I am presenting:

 

Pretty much what this notation is presenting is that using the Hailstone sequence, sum the reciprocals of all the numbers of the hailstone sequence, besides 1, to a get a solution.

 

My conjecture here is that the summation will always have a high bound 3(unsure about a lower bound). I have tested with many numbers and haven't found a counter example, though larger numbers may present different results.

 

Another part of the conjecture is if the solution can get close to 3, but never reach 3, then that means all sequences are finite.

Edited May 16, 2014 by Unity+