Approximation of $\left(\frac{n - 1}{n}\right) ^ n$

Question

Is there a good approximation function for

$$\left(\frac{n - 1}{n}\right) ^ n$$

I am trying to quickly calculate the probability of an event with probability of $\frac{1}{n}$ not happening if you try $n$ times.

It's quite hard to google something you can't put a name on.

Answer 1

It is $$\lim_{n\to \infty} \left(1-\frac{1}{n}\right)^n=e^{-1}$$

Answer 2

Note that

$$\left(\frac{n-1}n\right)^n=\left(1-\frac1n\right)^n=\frac{n-1}{1\times2n}-\frac{(n-1)(n-2)}{1\times2\times3n^2}+\frac{(n-1)(n-2)(n-3)}{1\times2\times3\times4n^3}+\dots$$

which is the binomial expansion theorem.

Answer 3

The function $$ f(x) = \left(\frac{x-1}x\right)^x=\left(1-\frac1x\right)^x $$ is well-behaved, it appears, and, as others have noted, it approaches $1/e$ as $x$ gets larger. Here is a graph of $1/e - f(x)$:

So you can fit the first part of it with a polynomial, which will speed up your calculations, and you can just use $1/e$ for large values of $x$. When $x=100$, $1/e -f(x) \approx 0.00185.$

An excellent tool for fitting with polynomials is chebfun.

Answer 4

If you want a better approximation than $\frac1e \approx 0.36787944$ then you could try the Laurent series

$$\left(\dfrac{n-1}{n}\right)^n = \frac1e\left(1 - \frac1{2x} - \frac5{24x^2}- \frac5{48x^3} - \frac{337}{5760x^4} -O\left(\frac1{x^5}\right) \right)$$

which for small $n$ and ignoring the $O\left(\frac1{x^5}\right)$ term gives

n       ((n-1)/n)^n Approx      difference
1       0           0.0474539   0.0474539
2       0.25        0.2506139   0.0006139
3       0.2962963   0.2963655   0.0000692
4       0.3164063   0.3164216   0.0000153
5       0.32768     0.3276848   0.0000048
6       0.3348980   0.3348999   0.0000019
7       0.3399167   0.3399175   0.0000009
8       0.3436089   0.3436094   0.0000004
9       0.3464394   0.3464397   0.0000002
10      0.3486784   0.3486786   0.0000001
11      0.3504939   0.3504940   0.0000001
12      0.3519956   0.3519957   0.0000001
13      0.3532585   0.3532585   0.0000000
14      0.3543353   0.3543353   0.0000000
15      0.3552644   0.3552644   0.0000000
16      0.3560741   0.3560741   0.0000000
17      0.3567862   0.3567862   0.0000000
18      0.3574172   0.3574172   0.0000000
19      0.3579803   0.3579803   0.0000000
20      0.3584859   0.3584859   0.0000000
21      0.3589424   0.3589424   0.0000000
22      0.3593565   0.3593565   0.0000000
23      0.3597340   0.3597340   0.0000000
24      0.3600794   0.3600794   0.0000000
25      0.3603967   0.3603967   0.0000000
100     0.3660323   0.3660323   0.0000000
1000    0.3676954   0.3676954   0.0000000