Given a sequence $(g_m)_{m\geq 1}$ with the asymptotic series $$g_m=\frac{a_1}{m} + \frac{a_2}{m^2} + \frac{R_m}{m^3}$$ where the numbers $a_1$, $a_2$ are known and we know $0<R_m<0.35$.
Given $Z$ with $|Z|<1$, the sequence $(f_n)_{n\geq 1}$ is defined by $$f_n=\sum_{k=0}^\infty Z^k\sqrt{\frac{n}{n+k}}\,\exp(g_{n+k}).$$
Question: How can I derive an asymptotic series of $f_n$ around $n=\infty$ with a bound on the remainder $r_n$? $$f_n=b_0+\frac{b_1}{n}+\frac{b_2}{n^2}+\frac{r_n}{n^3}$$
It would be even better to have $\ln(f_n)$ as a similar series with bound on the remainder. Any ideas?