Calculating Euler's Number

We are well aware of Euler’s famous number,

$$
e = 2.71828182845
$$

There are also quite a few ways of deriving Euler’s Number. There’s the Taylor expansion method:

$$
e = \sum _{k=0} ^{\infty} \frac{1}{k!}
$$

There is also the classical limit form:

$$
e = \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n} \right)^n
$$

Then there is another. Let $R$ be a random number generated between $[0, 1]$, inclusive. Then $e$ is the average of the number of $R$ s it takes to sum greater than $1$.

With more rigor, for uniform $(0,, 1)$ random independent variables $R_1$, $R_2$, $\ldots$, $R_n$,

$$
N = \min \left\{n: \sum_{k=1}^n R_k > 1 \right\}
$$

where

$$e = \text{Average}(n)$$

The proof can be found here, but it’s pretty math-heavy. Instead, an easier method is to write a program to verify for large enough $n$.

Source Code

def e_sum(upper_bound):
	if upper_bound < 1:
		return 0

	return Decimal(1.0) / 
	  Decimal(math.factorial(upper_bound)) +  \
      Decimal(e_sum(upper_bound - 1))

def e_limit(n):
	return Decimal((1 + 1.0 / float(n))**n)


def find_greater_than_one(value=0, attempts=0):
	if value <= 1:
		return find_greater_than_one(value + random.uniform(0, 1), attempts + 1)

	return attempts