I'm using a recursive function $f: \mathbb{N} \rightarrow \mathbb{N}$, that is defined as \begin{equation} f(n)=\lceil \log(f(n-1)) \rceil +f(n-1) \end{equation}

where $f(1)=F\in \mathbb{N}$, and $\lceil \cdot \rceil$ is the upper integer part of $\cdot$ (that is the cieling function). I would like to either find a close expression or an approximation to $f(n)$, that could help me finding the following limit (or an upper bound) \begin{equation} \lim_{N\rightarrow \infty } \sum_{n=2}^N \log(f(n)) \frac{\lceil \log(f(n)) \rceil - \lceil \log(f(n-1)) \rceil }{\lceil \log(f(n)) \rceil \cdot \lceil \log(f(n-1)) \rceil} \end{equation}

I believe that the limit should exists, since the numerator is almost always $0$ for large $n$. I will appreciate any help, or tip you could give me.

Thanks!