From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: I Need Proof: pi is Irrational
Date: 3 Feb 1999 15:49:26 -0500
Newsgroups: sci.math
Keywords: e is irrational
In article <7992d4$caa$1@sparcserver.lrz-muenchen.de>,
Helmut Richter wrote:
[concerning irrationality of e, and a failed attempt to prove it]
>As others have pointed out, this is a bit too skimpy as a proof, but I
>remember having seen a real proof along these lines. If someone is
>interested, I can try to find it.
It is faster and more elegant to prove the irrationality of 1/e:
the series (after obvious simplification) is
1/e = 1/2! - 1/3! +1/4! - ...
Now for every positive integer q >= 2, multiply by q! and separate:
q!/e = q!/2! - q!/3! + ... +(-1)^q * q!/q!
+ (-1)^(q+1) * (1/(q+1) - 1/((q+1)*(q+2)) + ...)
The expression in the first line is an integer, and the remainder is never
zero but is in absolute value less than 1, because of the error estimates
for alternating series (first neglected term...)
So, q! * 1/e is never an integer, and it follows that 1/e is not a
rational number (by contradiction). Hence e is also irrational.
Cheers, ZVK(Slavek).