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A remarkably elementary proof of the irrationality of e

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 Sn denote the nth partial sum of the series:

Notice that Sn 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].)

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