I have two sequences:
- $p_0, p_0 \rho_1, p_0 \rho_2, ...$
- $q_0, q_0 \gamma_1, q_0 \gamma_2, ...$.
Both sequences have infinite number of terms, and both sequences sum to 1 each. All terms are finite and non-negative. Further, I know that $\rho_1 \ge \rho_2 \ge \rho_3...$ and $\gamma_1 \ge \gamma_2 \ge \gamma_3...$, so all the terms in both sequences are decreasing. And, I know that $\rho_i < \gamma_i, \forall i$.
I want to say something about comparing the respective terms of the two series; i.e., how does the (say) third term of series 1 compare with the third term of series 2? It is hard to simply divide the two terms and check. Is there a smarter way?