Just to give credit where credit is due: Where I wrote "perhaps the first", that was entirely my wrong idea, not Jonathan Sondow's. Rich wrote: << Googling for 'e irrational' turns up a 2006 MAA article by Ed Sandifer that credits Euler in 1737 for proving the irrationality of e and e^2. The method is somewhat roundabout, proving the continued fractions from the Ricatti equation, and noting that the infinite CF -> irrational. My (admittedly rusty) memory is that the CF for tanh(x), and I1/I0, can be established directly by doing GCD-like steps on the power series for sinh/cosh; the subsequent steps in the GCD process have relatively clean power series. For full rigor, you'd need to prove some convergence results. But this makes the results a little less mysterious. I wrote: << I think I just found some article that claims this representation was invented by Cantor. I've read (in a Monthly article by Jonathan Sondow) that the fact Warren states below (which has a 0-line proof) was used by Cantor to give perhaps the first proof that e is irrational, as an immediate corollary.
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