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August 30th, 2016, 05:41 PM   #1
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Finding the bounds of a ratio

Sorry in advance if I've posted in the wrong section.

given the set $\{r_i, r_{ii}, r_{iii}, ... , r_R\}$
where $r_i \geq r_{i+1}$


What are the bounds of this ratio if it has any?

$$( \, log_2{\frac{(\sum_{i=1}^R{r_i})!}{\prod_{i=1}^R{(r _i!)}}}) \, :\sum_{i=1}^{R}{i \cdot r_i}$$

Last edited by iScience; August 30th, 2016 at 05:45 PM.
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August 30th, 2016, 06:35 PM   #2
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Quote:
Originally Posted by iScience View Post
Sorry in advance if I've posted in the wrong section.

given the set $\{r_i, r_{ii}, r_{iii}, ... , r_R\}$
where $r_i \geq r_{i+1}$


What are the bounds of this ratio if it has any?

$$( \, log_2{\frac{(\sum_{i=1}^R{r_i})!}{\prod_{i=1}^R{(r _i!)}}}) \, :\sum_{i=1}^{R}{i \cdot r_i}$$
I can't solve it myself but I'm guessing you are taking the r_i to be positive integers?

-Dan
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August 30th, 2016, 07:08 PM   #3
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oh oops. Yeah

$$r \ \epsilon \ \mathbb{Z}_+ $$

Last edited by iScience; August 30th, 2016 at 07:12 PM.
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September 1st, 2016, 10:43 PM   #4
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Basically the domain for this ratio is a $set$ of positive integers, bounded from $[1 , inf)$

My goal is find the range of that ratio, given the domain.
For all tested sets of values so far, I haven't found the range (value of the ratio) to ever reach 1. I can't even come up with a proof that this must be true for all cases. Therefore, continuing forward.. assuming that this is true (ratio < 1), I'd like to know what properties of a given set dictates the ratio's closeness to the value 1.


Any help is truly appreciated.
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