# Finding the bounds of a ratio

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

given the set ##\{r_i, r_{ii}, r_{iii}, ... , r_R\}##
where ##r \ \epsilon \ \mathbb{Z}_+ \ , \ r_i \geq r_{i+1}##

How would you go about finding the bounds of something like this, or determining if it even 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}##

Since ##r_i## is the largest and ##r_R## is the smallest, the ratio in the log must smaller than ##\frac{(r_iR)!}{(r_R!)^R},## which may depend on ##r_i##, ##r_R## and ##R.## Does the problem require a fixed constant?

Since ##r_i## is the largest and ##r_R## is the smallest, the ratio in the log must smaller than ##\frac{(r_iR)!}{(r_R!)^R},## which may depend on ##r_i##, ##r_R## and ##R.## Does the problem require a fixed constant?

The fraction within the log is always greater than or equal to one, and only gets larger and larger with increasing ##R## and ##\bar{r}##. But I don't know how I would go about expressing the rate of change with respect to the denominator of the ratio.
Hmm, I guess... what I'm looking for is more of a simplification, because that would be easier to look at. But I'm not sure what you meant by the fixed constant thing, can you elaborate?

At first I'm not sure what you want also, and guess maybe you hope to get the ratio can be bounded with some cool number, maybe ##Ce^R##... likewise haha.
For me the fraction cannot be simplified XD

For any set of values I've tested so far, the ratio has not exceeded the value 1. In fact i haven't found it to ever reach 1. I wanted to know what properties of a given set determined its closeness to the value 1. I hope this makes sense.

What if I choose ##r_i=100,r_R=1,R=2?## Then the ratio is greater than 1 right?

##\{100\ ,\ 1\}## yields...

##numerator: log_2{ \frac{ 101! }{ 100! \cdot 1! } } = log_2{ \frac{ 101! }{ 100! } } = log_2{101} =7##
##ratio: \frac{7}{102} = 0.0686##

(remember the numerator is logged by base 2)

...My fault to miss the latter. I just saw the fraction. So the sum of ##r_x## is also concerned... Then it's harder to simplify...

I know I stated that simplification was my goal. But I suppose that's not the only way to achieve my "true goal", which I'm still getting closer to.. (sorry)

I postulate that there is a relationship between the numbers in the given set, that dictates its closeness to the value 1. I just want to figure that out..
even a pointer to how I might go about that would be of great help.

Stephen Tashi