natural log base

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

$$2.718281828459045\mathrm{\dots }$$

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

$e$ is defined by

$$\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{1}{n}\right)}^n$$

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

$$e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\mathrm{\cdots }$$

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