The standard proofs of the ir rationality of e make use of the infinite series
representation

or the corresponding alternating series re presentation for 1/e . ( One such
proof is given at the end of this

article.) While these proofs are elementary, they obviously require some
familiarity with infinite series. The

fol lowing proof requires only integration-by-parts and some basic properties of
the Riemann integral. The

sum (1) follows as a consequence, thereby making this proof useful as an
introduction to infinite series.

e is irrational.

Proof: Suppose e = a/b, where a and b are positive integers . Choose an
integer n≥max{b, e}. Now

consider the definite integral
This integral is easily evaluated to give 1 − 1/e . On the other hand, repeated
integration-by-parts ( n times ) gives

Upon multiplying both sides by e and isolating the integral, we obtain

Multiplying both sides of (2) by n! gives

Because of the choice of n and the as sumption that e is rational , the left
hand side must reduce to an integer .

However the value of the expression on the right is between zero and one . Indeed

This contradiction implies that e must be irrational.

Notice that the integral in (2) approaches zero as
Therefore we obtain (1) as a by- product of the

proof . The series representation (1) was derived in a similar way by Chamberland
in [1] and by Johnson in

[2].

**A proof using the series for 1/e ...**

Use the fact that

and let S_{n} denote the nth partial sum of the series:

Notice that S_{n} is a rational number, and it can be written in the
form M/n!, where M is an integer. By the

alternating series estimation theorem, it follows that

In either case, e^{−1} is strictly between two
rational numbers of the forms , where a is an

integer. It follows that e^{−1} cannot be written as a fraction with denominator (n
+ 1)! for any n≥0. Since

any rational number can be written as a fraction with denominator (n+1)!, we
conclude that e^{−1} cannot be

a rational number. Since 1/e is irrational, it follows that e is irrational.
(This proof is similar to Sondow’s

geometric proof [3].)