__I. The Collatz Function as a Limit at Infinity__

One of the reasons, which is not often discussed, for the convergence of the Collatz function

^{[1]}is that it relies upon indefinite division by fluctuating powers of two. In otherwords, if we are only allowed to divide an even number $n$ by $2^k$ for some maximal $k$ (regardless of whether said number is divisible by a greater power of two), the resulting sequence will diverge to infinity for most values of $n$. More formally, we can represent the entire family of*restricted Collatz functions*as\[ f_k(n) = \begin{cases} 3n+1, & n \equiv 1 \ (\text{mod} \ 2) \\ n-2, & n \equiv 0 \ (\text{mod} \ 2^{k+1}) \\ n/2, & \text{otherwise} \end{cases} \]

for $k \geq 1$.