Let $\gamma>0$ and $\{a_n\}_{n>0}$ and $\{b_n\}_{n>0}$ be sequences defined as follows $$ a_n := \sum_{k=0}^{n-1}{2k \choose k}\frac{1}{\gamma^{2k+1}} \left(1-e^{-\frac{\gamma}{n}}\sum_{h=0}^{2k} \frac{(\gamma/n)^h}{h!} \right) $$ $$ b_n := \sum_{k=0}^{n-1}{2k \choose k}\frac{1}{\gamma^{2k+1}} e^{-\frac{\gamma}{n}}\sum_{h=0}^{2k} \frac{(\gamma/n)^h}{h!}. $$ Moreover, let $$ c_n:=\left(\frac{a_n}{b_n}\right)^n, \quad n>0. $$
My question. Does the sequence $\{c_n\}_{n>0}$ converge? If so, to which value?
Note. This is not an homework. It is part of a problem that came up in my research. Experimentally, I noticed the $\{c_n\}_{n>0}$ converges to a finite non-zero value for any randomly selected $\gamma>0$. Thus, I wonder whether this can be formally proved.