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Problem D
Stirling's Approximation
The value of $n! = 1\cdot 2 \cdot 3 \cdot \ldots \cdot (n-1) \cdot n$ can be quite large, and it can be tedious to calculate. Fortunately for us, the French mathematician Abraham de Moivre (1667-1754) and the Scottish mathematician James Stirling (1692-1770) came up with a nice approximation:
\[ n!\approx S(n)=\sqrt {2\pi n}\frac{n^ n}{e^ n}. \]This is known as Stirling’s approximation.
For this problem, determine how good Stirling’s approximation is for various $n$.
Input
Input consists of up to $500$ test cases, one per line. Each test case has a single integer $0 < n \leq 10^5$. Input ends when $n=0$.
Output
For each test case, print out $n! / S(n)$ with an absolute error at most $10^{-8}$.
| Sample Input 1 | Sample Output 1 |
|---|---|
1 10 150 0 |
1.084437551419227546611577313 1.008365359132400245905553271 1.000555709081632955425046604 |
