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Statement

Prove that Euler's number

e = \sum_{n=0}^{\infty} \dfrac{1}{n!}

is irrational.

Current frontier

Classical result (Fourier, 1815; the proof attributed to Fourier is the standard textbook one). Standard route: assume

e = p/q

for positive integers

p, q

, multiply both sides by

q!

to get

q!\, e = q!\, p / q

as a positive integer, split the series into the head

\sum_{n=0}^{q} q!/n!

(integer) and the tail

\sum_{n=q+1}^{\infty} q!/n!

, then show the tail strictly lies in

(0, 1)

— contradiction. Liouville's 1844 theorem later upgraded this to transcendence, but only irrationality is asked here.

When this counts as solved

A complete, self-contained proof that

e = \sum_{n=0}^{\infty} \frac{1}{n!}

is irrational. The argument has to derive everything it uses: the standard route assumes

e = p/q

, multiplies by

q!

, and bounds the resulting series-tail strictly between

0

and

1

to contradict integrality — every step there needs to be justified (in particular the tail bound

\sum_{n=q+1}^{\infty} \frac{q!}{n!} < \frac{1}{q}

must be derived, not asserted). Citing transcendence of

e

or Liouville's theorem to dodge the irrationality argument doesn't count. There's no partial credit on this one; only a finished proof closes it out.

Classification

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