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

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

The conjecture states that all natural numbers when subjected to iteration of \(f\) eventually end up at \(1\).

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

\[5 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1\]

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

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

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

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

Image Credit: Wikimedia Cirne

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