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Given two real numbers $0<a<1$ and $0<\delta<1$, I want to find a positive integer $i$ (it is better to a smaller $i$) such that $$\frac{a^i}{i!} \le \delta.$$

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Are you sure you want the minimum $i$ which will be hard to compute exactly? Or would you be happy with an $N$ such that any $i$ greater than $N$ would work (which is what you would need if you were looking at problem with limits). – Mark Bennet May 14 '12 at 11:15
If the minimum $i$ is too hard to find, then it is better to find an integer $i$ as small as possible – John Smith May 14 '12 at 11:18
@draks, how to use Lambert-W-function? I am now obtaining the following inequality: $${(a\cdot e)}^i<\sqrt{2\pi}\delta i^{i+1/2}$$ – John Smith May 14 '12 at 13:41
up vote 1 down vote accepted

Here is what I put together from my math toy box:

  1. Use Stirling's approximation $i!\approx(i/e)^i$ to get $\left( \frac{ae}{i}\right)^i \le \delta$.
  2. Call $ae=1/b$ and invert to get $(ib)^i\ge \delta^{-1}$.
  3. Continue with $(ib)^{ib}\ge \delta^{-b}$, define $x:=bi$ to get $x^x\ge\delta^{-b}$
  4. and then use $$ x\ge\frac{\ln(\delta^{-b})}{W(\ln \delta^{-b})}=\frac{\ln(\delta^{-1/ae})}{W(\ln \delta^{-1/ae})}. $$
  5. Resubstitute $x=\frac{i}{ae}$ for the result $i\ge\frac{ae\ln(\delta^{-1/ae})}{W(\ln \delta^{-1/ae})}$.
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How to estimate $W(\cdot)$? – John Smith May 14 '12 at 15:03
@JohnSmith, would the Asymptotic expansion $W_0 (x) = \sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!}\ x^n = x - x^2 + \frac{3}{2}x^3 - \frac{8}{3}x^4 + \frac{125}{24}x^5 - \cdots $ work for you? – draks ... May 14 '12 at 15:05
Also, $i! \sim {(i/e)}^i$ doesn't mean $i! \ge {(i/e)}^i$. This problem can be easily fixed by using the upper bound of $i!$ of Stirling Approx., but how to calculate $W(\cdot)$ is a very important issue. – John Smith May 14 '12 at 15:06
I think it's complicated to compute $W_0(x)$. – John Smith May 14 '12 at 15:11
Ask your favorite math knowlegde base and get this: $\displaystyle \log x - \log \log x + \frac{1}{2}\frac{{\log \log x}}{{\log x}} \le W(x) \le \log x - \log \log x + \frac{e}{{e - 1}}\frac{{\log \log x}}{{\log x}}$. Does that help? – draks ... May 16 '12 at 10:15

Here is a not-very-good answer. Let $i$ be the result of rounding $\log\delta/\log a$ up to the nearest integer. Then $i\ge\log\delta/\log a$, so $i\log a\le\log\delta$ (remember, $\log a\lt0$), so $a^i\le\delta$, so $a^i/i!\le\delta$.

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$\frac{a^i}{i!}<{a^i}$ seems too loose. What if Stirling's Approx. is applied? – John Smith May 14 '12 at 13:38
@JohnSmith: If you are programming, you can now just count downward in $i$ and check. – Ross Millikan May 14 '12 at 14:01
@JohnSmith: Stirling doesn't buy you much here: you will get i both in the base and in the exponent. – Johannes Kloos May 14 '12 at 14:07

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