proof that e is not a natural number
Here, we are going to show that the natural log base is not a natural number by showing a sharper result: that is between and .
We chop up the Taylor expansion of into two parts: the first part consists of the sum of the first two terms, and the second part consists of the sum of the rest, or . The proof of the proposition now lies in the estimation of and .
Step 2: e3. This step is the same as showing that . With this in mind, let us compare term by term of the series (2) representing and another series (3):
It is well-known that the second series (a geometric series) sums to 1. Because both series are convergent, the term-by-term comparisons make sense. Except for the first term, where , we have for all other terms. The inequality , for a positive number can be translated into the basic inequality , the proof of which, based on mathematical induction, can be found here (http://planetmath.org/AnExampleOfMathematicalInduction).
Because the term comparisons show
that the terms from (2) the corresponding terms from (3), and
that at least one term from (2) than the corresponding term from (3),
we conclude that (2) (3), or that . This concludes the proof. ∎
|Title||proof that e is not a natural number|
|Date of creation||2013-03-22 15:39:52|
|Last modified on||2013-03-22 15:39:52|
|Last modified by||CWoo (3771)|