# 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?

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

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