natural log base

The natural log base, or $e$, has value

 $2.718281828459045\ldots$

$e$ was extensively studied by Euler in the 1720’s, but it was originally discovered by John Napier.

$e$ is defined by

 $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n}$

It is more effectively calculated, however, by using the Taylor series for $f(x)=e^{x}$ at $x=1$ to get the representation

 $e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots$
 Title natural log base Canonical name NaturalLogBase Date of creation 2013-03-22 11:55:56 Last modified on 2013-03-22 11:55:56 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 11 Author CWoo (3771) Entry type Definition Classification msc 33B99 Synonym Euler number Synonym Eulerian number Synonym Napier’s constant Synonym e Related topic ExampleOfTaylorPolynomialsForTheExponentialFunction Related topic EIsTranscendental Related topic EIsIrrationalProof Related topic ApplicationOfCauchyCriterionForConvergence