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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?

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  • 2
    $\begingroup$ 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? $\endgroup$ – Robert Israel Feb 18 at 2:05
  • $\begingroup$ Couldn't we say something by looking at the rate of decay somehow? $\endgroup$ – bissi Feb 19 at 0:07
  • $\begingroup$ It could be. What do you know about the rates of decay? You certainly haven't told us anything about them. $\endgroup$ – Robert Israel Feb 19 at 3:51
  • $\begingroup$ added a comment now, does it help? $\endgroup$ – bissi Feb 19 at 21:14

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