TITLE: e is irrational
QUESTION [17 upvotes]: Prove that e is an irrational number.
Recall that $\,\mathrm{e}=\displaystyle\sum_{n=0}^\infty\frac{1}{n!},\,\,$ and assume $\,\mathrm{e}\,$ is rational, then
$$\sum\limits_{k=0}^\infty \frac{1}{k!} = \frac{a}{b},\quad \text{for some positive integers}\,\,\, a,b.$$
so
$$b\sum\limits_{k=0}^\infty \frac{1}{k!} =a$$
or
$$
b\left(1+1+\frac{1}{2} + \frac{1}{6} +\cdots \right)= a.
$$
Where can I go from here?

REPLY [2 votes]: Reading Courant-Robbins I was surprised to see that proving the irrationality of
$$
e = \sum_{n=0}^{\infty} \frac{1}{n!} = 2 + \frac{1}{2} + \frac{1}{2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} + \dots
$$
isn't harder than proving the irrationality of $\sqrt{2}$, just a bit more laborious (I fear that with $\pi$ things go in a very different way...). I found Courant-Robbins proof development very concise, so I developed the proof in 15 numbered steps (with the purpose to be more explicit and clear). In the first part of the proof (step $1\to 4$) we'll show that surely the first ten digit of $e$ are $2.718281828$ (for the second part of the proof it would be sufficient to prove that $e$ is not natural, but since we're playing, better do a good estimate). In the second part (step $5\to 15$) we'll focus on irrationality of $e$.

1 ) We start proving a formula that we will use two times (step 4 and step 14): if $k$ is a natural number bigger than one, we have
$$
\sum_{n=0}^{\infty} \frac{1}{k^n} = \frac{1}{1-\frac{1}{k}}
$$
To prove this, let's consider the sum
$$
G_n = q^0 + q^1 + q^2 + \ldots + q^n
$$
Multiplying both member by $q$ we get
$$
q G_n = q^1 + q^2 + q^3 + \ldots + q^{n+1}
$$
and subtracting we get (suppose $q \neq 1$) 
$$
G_n = \frac{1-q^{n+1}}{1-q}
$$
If $q=\frac{1}{k}$ with $k$ natural bigger than one, and if the sum goes to $n \to \infty$, the second term in the numerator goes to zero, qed.
2 ) Summing first 13 terms of $\sum \frac{1}{n!}$ (i.e. from $n=0$ to $n=12$ inclusive) we find
$$ \sum_{n=0}^{12} \frac{1}{n!} = \frac{260412269}{95800320}$$
that is about $2.71828182829$.
3 ) The difference $\xi$ between this estimate and the true value of $e$ is
$$
\xi = \frac{1}{13!} + \frac{1}{14!} + \frac{1}{15!} + \ldots = \frac{1}{13!} + \frac{1}{13!\cdot 14} + \frac{1}{13!\cdot 14 \cdot 15} + \ldots
$$
We can overvalue in this way
$$
\xi < \frac{1}{13!} + \frac{1}{13!\cdot 13} + \frac{1}{13!\cdot 13^2} + \ldots = \frac{1}{13!} \left( 1 + \frac{1}{13} + \frac{1}{13^2} + \ldots \right) 
$$
So
$$
\xi < \frac{1}{13!} \sum_{n=0}^{\infty} \frac{1}{13^n}
$$
4 ) Using the formula proved in step 1 we have 
$$
\xi < \frac{1}{13!} \cdot \frac{1}{1-\frac{1}{13}} = \frac{1}{12! \cdot 12} = \frac{1}{5748019200} \approx 1,7 \cdot 10^{-10}
$$
This means that the error due to the fact we stopped the sum to thirteenth term can influence at most the tenth digit after the dot (adding $0.00000000017$ to $2.71828182829$ we cannot change the first ten digit of the number). The first part of proof is ended: $e$ "starts" with $2.718281828$.
5 ) Let's prove the irrationality of $e$ by supposing, by absurd, that it is rational:
$$
e = \frac{p}{q} \qquad p,q \in \mathbb{N}
$$
6 ) We already proved that $e \approx 2.7$. In particular, $e \notin \mathbb{N}$, so
$$
q \ge 2
$$
7 ) Let's substitute in $e=\frac{p}{q}$ the definition of $e$ and let's multiply two member of this equation by $q!$ (on the right, I simplified $\frac{q!}{q}$ to $(q-1)!$)
$$
\left( \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots \right) q! = p (q-1) !
$$
8 ) Let's rewrite the first member in a different way 
$$
\begin{array}{c}
\displaystyle \left( 2 + \frac{1}{2!} + \frac{1}{3!} + \ldots \right. \\
\displaystyle \left. + \frac{1}{(q-2) !} + \frac{1}{(q-1) !} + \frac{1}{q!} + \frac{1}{(q+1) !} + \frac{1}{(q+2) !} + \ldots \right) q!
\end{array}
$$
Doing the product we have
$$
2 q! + 3 \cdot 4 \cdot \ldots \cdot q + 4 \cdot 5 \cdot \ldots \cdot q + \ldots + (q-1) \cdot q + q + 1 + \frac{1}{q+1} + \frac{1}{(q+1) (q+2) } + \ldots
$$
9 ) Let's define the number $A$, $B$ and $C$ in this way
$$
A = p (q-1) !
$$
$$
B = 2 q! + 3 \cdot 4 \cdot \ldots \cdot q + 4 \cdot 5 \cdot \ldots \cdot q + \ldots + (q-1) \cdot q + q + 1
$$
$$
C = \frac{1}{q+1} + \frac{1}{(q+1) (q+2) } + \ldots
$$
10 ) Using the number $A$, $B$ and $C$, and exploiting step 8, the equation of step 7 can be written in this way
$$
B + C = A
$$
11 ) $A$ e $B$ are clearly integer by definition, so if we prove that $C$ is not integer we have an absurdity starting by the hypothesis of rationality of $e$ and the proof is ended. 
12 ) The terms of the serie $\frac{1}{q+1} + \frac{1}{(q+1) (q+2) } + \ldots$ must be smaller (with the only exception of the first term) to the terms of the serie
$$
\frac{1}{q+1} + \frac{1}{(q+1) ^2} + \ldots = \sum_{n=1}^{\infty} \frac{1}{(q+1) ^n} = \sum_{n=0}^{\infty} \frac{1}{(q+1) ^n} - 1
$$
So we certainly have
$$
C < \sum_{n=0}^{\infty} \frac{1}{(q+1) ^n} - 1
$$
13 ) The serie $\sum_{n=0}^{\infty} \frac{1}{(q+1) ^n}$ take the greatest possible value when the natural number $q$ take the smallest possible value. So we certainly have ( $q=1$ is excluded, as showed in step 6)
$$
C < \sum_{n=0}^{\infty} \frac{1}{3^n} - 1
$$
14 ) Exploiting the theorem proved in the step 1, the last inequality becomes
$$
C < \frac{1}{1-\frac{1}{3}} - 1 = \frac{1}{2}
$$
15 ) We proved that $C<\frac{1}{2}$, so $C \notin \mathbb{N}$. Observing the step 11 we see that the proof is ended: $e$ is irrational.