3
$\begingroup$

In previous question, I asked how one would simplify the following equation for the case where the variables are very big:

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

This answer was basically to use an approximation like Stirling's formula. Having implemented this with some code, it still takes too long to find the maximum value of N so that the inequality holds true. Therefore, I need a direct solution for N. So the new question is, how would you go about solving this equation for N?

(Some simplifications are acceptable, but I would like to have it as accurate as possible. The values this equation will be used for are all in the 100,000-1,000,000 range, except $m$, which is in the 100s range.)

  • 0
    What can be said about *m* and *k*? Because if *m* were smal, the sum with *m* = 0 can be probably computed explicitely using exponential generating functions (e.g.f.) and then you can approximate only the first terms. Of course, if *m* is too big, this is not feasible. Also if *k* is too big, the solution with e.g.f. might not help you much.2010-11-12

1 Answers 1