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August 30th, 2016, 06:41 PM  #1 
Newbie Joined: Jan 2014 Posts: 16 Thanks: 0  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 06:45 PM. 
August 30th, 2016, 07:35 PM  #2  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,345 Thanks: 986 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  
August 30th, 2016, 08:08 PM  #3 
Newbie Joined: Jan 2014 Posts: 16 Thanks: 0 
oh oops. Yeah $$r \ \epsilon \ \mathbb{Z}_+ $$ Last edited by iScience; August 30th, 2016 at 08:12 PM. 
September 1st, 2016, 11:43 PM  #4 
Newbie Joined: Jan 2014 Posts: 16 Thanks: 0 
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. 

Tags 
analysis, bounds, discrete, discrete math, finding, ratio 
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