Collapsing from Collatz (300 followers problem!)

Calculus Level 4

The 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."

Consider the function

f(n)={n/2if n is even3n+1if n is oddf(n) = \begin{cases} n/2 & \text{if } n \text{ is even} \\ 3n+1 &\text{if } n \text{ is odd} \end{cases}

The conjecture states that all natural numbers when subjected to iteration of ff eventually end up at 11.

Let the (minimum) number of iterations of ff taken by nn be (n)\ell(n). For example,

51684215 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1

Hence, (5)=5\ell(5) = 5.

If nn never reaches 11, then (n)=\ell(n) = \infty, with the interpretation that 1(n)=0 \frac{1}{\ell(n)} = 0 .

What can be said about the convergence of the following series?

n=21(n)\sum_{n=2}^{\infty} \frac{1}{\ell(n)}

Image Credit: Wikimedia Cirne

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