# How to address an inequality having the amount of factorials and also powers

In previous question, I asked just how one would certainly streamline the list below formula for the instance where the variables are large:

$\sum\limits^{k}_{i=m}(N-i)^{k-i}(\frac{1}{N})^k\frac{k!}{(k-i)!i!} \leq a$

This solution was primarily to make use of an estimate like Stirling is formula. Having actually applied this with some code, it still takes also lengthy to locate the maximum value of N to make sure that the inequality applies. Consequently, I require a straight remedy for N. So the new inquiry is, just how would certainly you deal with addressing this formula for N?

(Some simplifications serve, yet I would love to have it as exact as feasible. The values this formula will certainly be made use of for are done in the 100,000 - 1,000,000 array, other than $m$, which remains in the 100s array.)

Now you are primarily right into origin searching for. I enjoy Numerical Recipes. I have had guide over 30 years. Excellent conversation of the strategies and also code provided in numerous languages. Your formula needs to be a very easy one. Various other mathematical evaluation publications will certainly function, also.

A tiny idea: if m is as tiny as you claim, there could be some algebraic simplification in making it 1. I have not located it, yet others are much better at these combinatoric identifications.

If you are in fact doing the amount every single time, you could be able to recognize the series of i that is the large payment. It resembles it needs to be rather listed below k/2. You can possibly lower the terms in the amount by a huge variable, at the very least to get near N. After that make use of the complete formula for last improvement. And also take the N ^ (- k) out of the amount.

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