Talk:Proof that e is irrational
|
|
I think the proof should also prove that <math>\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=\sum_{n=0}^{\infty}\frac{1}{n!}<math> 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)
Proofs of the equivalence of the definitions of e
See Definitions of the exponential function.
Brianjd 07:44, Jul 26, 2004 (UTC)