Limit of a recursive function

Question

I have a recursive functoin:

$$\log^{'}(n) = \begin{cases} & 1 \text{ if } n \leqslant 1 \\ & 1 + log^{'}(\log(n))\text{ otherwise} \end{cases}$$

This function grows VERY slowly. Is this function limited? How do I compare: $$n^2 \;\;\text{and}\;\;\; 2^{\log^{'}(n)}$$ Is it true that $n^2$ grows slower than $2^{\log^{'}(n)}$ for $n\rightarrow \infty$?

Answer 1

Compare the order of the functions you have, set $\log n =t$ and $\log t =v$ and take the limit of the ratio. What do you get?