Talk:Proof that e is irrational

I think the proof should also prove that $\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}=\sum _{n=0}^{\infty }{\frac {1}{n!}}$ given that e is defined as the left hand side expression. -- Dissident 05:42, 27 Feb 2004 (UTC)

Well, there are many ways of defining e, and since they all define the same number, no one definition is "the" definition of e. You can define it by the compound interest definition; I can define it by the infinite series, neither of us is "right" or "wrong". We just have to prove between the 2 of us that our definitions agree. What is proved here is that the number e, as defined by the series definition, is irrational. Then you can prove that any of the other definitions of e are irrational, by proving that all the definitions themselves give the same number. But that's a different problem, it seems to me. Part of the utility and beauty of having so many different equivalent definitions for e is that you can choose the one that best suits your needs in a particular problem.

In order to do what you want to do, the only way I can see to do it, (off the top of my head) would be to look at the exponential function e(x) as defined by (1 + (x/n))ⁿ as n goes to infinity, take the derivative of this function with respect to x, and show that you get the same function. Then, you can calculate the Taylor series of the function at x = 0 by noting that all the derivatives are equal to 1, then plug in x = 1 and note that the two expressions are in fact equal. But to do this really goes off the beaten track, for one thing, you have to interchange the limit and the derivative, which has to be justified, then you have to derive the formula for Taylor series, etc, etc. It doesn't seem to have much to do with why e is irrational. Revolver 06:09, 27 Feb 2004 (UTC)

Just to give you an idea how many ways there are to "define" e, this is also a definition frequently used in analysis books:

• Define the natural logarithm function log(x) for all x > 1, by integrating dt/t from 1 to x, and prove this is strictly increasing. Then define e to be the unique number x > 1 such that log(x) = 1. Revolver 06:14, 27 Feb 2004 (UTC)

I found a relatively straightforward proof of the identity you give, that doesn't involve derivatives or Taylor expansions, just some inequalities and playing with lim sups...it's in baby Rudin ("Principles of mathematical analysis") in the chapter on sequences and series. But my opinion is still the same -- e.g. baby Rudin defines e by the series expansion. Revolver 02:45, 29 Feb 2004 (UTC)

Can you show it here? I'm pretty interested in it and it might be useful for Wikipedia. -- Dissident 19:33, 1 Mar 2004 (UTC)

I think there is a problem with this proof. Defining e as an infinite sum where all intermediate terms are rational means that using any intermediate (i.e. finite) result in the proof is the equivalent of using a rational approximation of e.

Another shortcoming of this proof is that there is no explanation as to why defining x in the way shown allows any conclusions about e itself to be drawn. 11/17/05 gwj

There is no shortcoming. This is a simple proof by contradiction. The definition of x includes e expressed as a rational number; when this results in a contradiction, the supposition that e is rational must clearly be false. —Lowellian (reply) 05:02, 6 January 2006 (UTC)

Proofs of the equivalence of the definitions of e

See Definitions of the exponential function.

Brianjd 07:44, Jul 26, 2004 (UTC) how do u used e= a/b to change e= sum to infinity of the series 1/n! to x= b!(e-sum from 0 to b of 1/n!)?

The statement 'Since this continued fraction is infinite and every rational number has a terminating continued fraction, e is irrational' is not logically sound. We need a reason why the fact that this object (e) has property 1 (an infinite continued fraction representation) prevents it from having property 2 (a terminating continued fraction representation) and thus from belonging to set in question (the rationals). There seems to be a tacit assumption that continued fraction representations of real numbers are unique, or that, if they are not, at least all the continued fractions representations of the same real number are of the same type (either all terminating or all infinite). This assumption may be correct for the case at hand (the continued fractions representations of a real number), but without this knowledge, the logical argument is incomplete. — Preceding unsigned comment added by 2601:546:C400:2F70:7998:5E6B:2009:A0DF (talk) 16:29, 8 March 2020 (UTC)

Infinite sum

The formula for the infinite sum of a geometric series is used, but incorrectly, as the sum used counts only when summed from 0 to infinity, where as in this case, it is a sum from 1 to infinity. That means 0 < x < 1+ 1/b, which ofcourse makes the proof inadequate.

There is no sum from 1 to infinity here! Oli Filth 17:55, 6 August 2007 (UTC)

Ah, I see, sorry, my mistake. :) Thanks!

x < 1

Shouldn't we prove also that x < 1? It's straightforward, but I think it should be there for the sake of completeness

I don't see how it's straightforward at all. I see nothing from this proof that implies that x < 1. Can someone explain? All I can see is that b must be greater than 1 since e isn't an integer. That shows that x < 1, but it doesn't seem very rigorous.

—Preceding unsigned comment added by 76.182.194.195 (talk) 12:34, 14 March 2010 (UTC)

In the formula x=sum b!/n!, the ith term is less than 1/2ⁱ, so the whole sum is less than 1/2+1/4+1/8+...=1. —David Eppstein (talk) 17:31, 14 March 2010 (UTC)