Is there a general procedure for evaluating series of the form:
$$\sum_{t=0}^\infty \frac{a_1 p_1^t+\cdots+a_m p_m^t}{b_1 q_1^t+\cdots+b_n q_n^t},$$
where $a_1,\dots,a_m,p_1,\dots,p_m,b_1,\dots,b_n,q_1,\dots,q_n\in\mathbb{R}^+$ and $m,n\in\mathbb{N}$?
You may assume that the series converges.