This article covers much about the mathematical constant e, Euler's number, concluding with the result that it is irrational.
The mathematical constant e was first found by Bernoulli with the formula
We will use this formula to determine a new formula for e and then we will use it to prove e's irrationality.
Lemma 1. The sequence increases.
Proof. We need to show
which is equivalent to
Simplifying, we get
If we change parameters and set , we get
which simplifies to
which, of course, holds.
Lemma 2. The sequence has an upper bound of .
Proof. It follows directly:
Corollary 3. The sequence converges.
Proof. Follows directly from Lemmas 1 and 2.
Lemma 4. The expression is equal to .
Proof. Since the first expression is equal to due to the Bolzano-Weierstrass theorem, it immediately follows.
Proof. It is obvious:
Lemma 6. The Taylor series expansion for is
Proof. We know an expression for , so we will differentiate it to obtain a result. It is obvious that this expression is 1 when k=0, so if we make the definition , . Now we differentiate our expression.
As we can see, , so all of the other derivatives will be 1 when evaluated at 0. This yields the following Taylor series for our function:
And the proof is complete.
Corollary 7. An infinite sum representation of e is
Proof. We take x=1 in the previous lemma to obtain this.
The proof of the theorem
We will use a proof by contradiction. Let's assume e is rational and for , it can be written as p/q. Observe the following equalities:
Since the expression on the RHS is a positive integer, so must be the expression on the LHS. This yields:
The last term of the last sum must be an integer because the sum equals a positive integer and its first term is a positive integer. This is important, the contradiction follows from here. Let's call this integer R. Now, observe these:
Now also note that q > 0, so all the terms in R are strictly positive, therefore R > 0. So we have 0 < R < 1, but we earlier established that R was a positive integer. As there are no integers between 0 and 1, we have a contradiction. Hence, it is impossible to express e as a ratio of two integers, so it must be irrational. Proof complete.