# Collapsing from Collatz (300 followers problem!)

**Calculus**Level 4

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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