Famous Theorems of Mathematics/e is irrational

The series representation of Euler's number e


 * $$e = \sum_{n = 0}^{\infty} \frac{1}{n!}\!$$

can be used to prove that e is irrational. Of the many representations of e, this is the Taylor series for the exponential function ey evaluated at y = 1.

Summary of the proof
This is a proof by contradiction. Initially e is assumed to be a rational number of the form a/b. We then analyze a blown-up difference x of the series representing e and its strictly smaller bth partial sum, which approximates the limiting value e. By choosing the magnifying factor to be b!, the fraction a/b and the bth partial sum are turned into integers, hence x must be a positive integer. However, the fast convergence of the series representation implies that the magnified approximation error x is still strictly smaller than 1. From this contradiction we deduce that e is irrational.

Proof
Suppose that e is a rational number. Then there exist positive integers a and b such that e = a/b.

Define the number


 * $$\ x = b!\,\biggl(e - \sum_{n = 0}^{b} \frac{1}{n!}\biggr)\!$$

To see that x is an integer, substitute e = a/b into this definition to obtain



x = b!\,\biggl(\frac{a}{b} - \sum_{n = 0}^{b} \frac{1}{n!}\biggr) = a(b - 1)! - \sum_{n = 0}^{b} \frac{b!}{n!}\,. $$

The first term is an integer, and every fraction in the sum is an integer since n≤b for each term. Therefore x is an integer.

We now prove that 0 < x < 1. First, insert the above series representation of e into the definition of x to obtain


 * $$x = \sum_{n = b+1}^{\infty} \frac{b!}{n!}>0\,.\!$$

For all terms with n ≥ b + 1 we have the upper estimate
 * $$\frac{b!}{n!}

=\frac1{(b+1)(b+2)\cdots(b+(n-b))} \le\frac1{(b+1)^{n-b}}\,,\! $$ which is even strict for every n ≥ b + 2. Changing the index of summation to k = n – b and using the formula for the infinite geometric series, we obtain

x =\sum_{n = b+1}^{\infty} \frac{b!}{n!} < \sum_{k=1}^\infty\frac1{(b+1)^k} =\frac{1}{b+1}\biggl(\frac1{1-\frac1{b+1}}\biggr) = \frac{1}{b} \le 1. $$

Since there is no integer strictly between 0 and 1, we have reached a contradiction, and so e must be irrational.