Given:

Presume for contradiction that e is rational, therefore

Such that p and q are integers. Since q is an integer there must be the term in the series of e, thus

Multiply both sides by q to get

The term p(q-1)! must be an integer as is since q! is divisible by all factorials including q!.

It follows that

The left side is greater than zero. The right side

is an infinite series with

Therefore,

demonstrating that is not an integer. Thus

for integer p and q. Therefore e is irrational. Q.E.D.

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