# comparing two decaying sequences

I have two sequences:

1. $$p_0, p_0 \rho_1, p_0 \rho_2, ...$$

and

1. $$q_0, q_0 \gamma_1, q_0 \gamma_2, ...$$.

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?

• Without knowing more about these sequences, it's pretty much impossible for us to make a sensible suggestion. You say it's "hard" to divide and check, but what is not hard? – Robert Israel Feb 18 at 2:05
• Couldn't we say something by looking at the rate of decay somehow? – bissi Feb 19 at 0:07
• It could be. What do you know about the rates of decay? You certainly haven't told us anything about them. – Robert Israel Feb 19 at 3:51
• added a comment now, does it help? – bissi Feb 19 at 21:14