Collatz conjecture (CC) is a well-known unsolved problem in mathematics, first proposed in 1937 by Lothar Collatz. Because of its overwhelming simplicity, the conjecture has been attacked many times - never successfully. The great Paul Erdős himself admitted: "Mathematics may not be ready for such problems."The
Consider the function
The conjecture states that all natural numbers when subjected to iteration of eventually end up at .
Let the (minimum) number of iterations of taken by be . For example,
If never reaches , then , with the interpretation that .
What can be said about the convergence of the following series?