I have the following hideous equation $$ \frac{1}{n} \log_q (q^lx) $$ with \begin{equation} x=\sum_{i=0}^{\lfloor \frac{n}{l} \rfloor -1} \left( \sum_{j=0}^{\lfloor \frac{n-(i+1)l}{l} \rfloor} (1-q)^j q^{n-(i+1)l-(l+1)j} \binom{n-(1+i+j)l}{j} - \sum_{j=0}^{\lfloor \frac{n-(i+1)l-l}{l} \rfloor} (1-q)^j q^{n-(i+1)l-l-(l+1)j} \binom{n-(2+i+j)l}{j} \right) \end{equation} with $1 \le l \le n$ and $ 2 \le q $.
I am interested in a analytical equation for the asymptotical behaviour with $n \rightarrow \infty$. $l$ and $q$ will have small values. Can anyone give me some hints, how to get a nice equation for $n \rightarrow \infty$?
Numerical simulations show that this equation approaches to the value $0.6958$ for the parameters $q=2,l=2$ and $n \rightarrow \infty$.