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Difference between revisions of "E-number"

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The limit of the expression $(1+1/n)^n$ as $n$ tends to infinity:
 
The limit of the expression $(1+1/n)^n$ as $n$ tends to infinity:
  
$$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n=2.718281828459045\ldots;$$
+
$$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n=2.718281828459045\dots;$$
  
it is the base for the natural logarithm. $e$ is a transcendental number, which was proved by C. Hermite in 1873 for the first time. Sometimes $e$ is called the Napier number for no very good reason.
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it is the base for the natural logarithm. $e$ is a transcendental number, which was proved by C. Hermite in 1873 for the first time.  
  
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$e$ is also defined as the sum of the series
  
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$$\sum _{n=0} ^{\infty} \frac{1}{n!}$$
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 +
That means
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 +
$$e=\sum _{n=0} ^{\infty} \frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\dots$$
  
 
====Comments====
 
====Comments====
 
See also [[Exponential function|Exponential function]]; [[Exponential function, real|Exponential function, real]]; [[Logarithm of a number|Logarithm of a number]]; [[Logarithmic function|Logarithmic function]]; [[Transcendental number|Transcendental number]].
 
See also [[Exponential function|Exponential function]]; [[Exponential function, real|Exponential function, real]]; [[Logarithm of a number|Logarithm of a number]]; [[Logarithmic function|Logarithmic function]]; [[Transcendental number|Transcendental number]].

Latest revision as of 15:10, 30 December 2018

The limit of the expression $(1+1/n)^n$ as $n$ tends to infinity:

$$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n=2.718281828459045\dots;$$

it is the base for the natural logarithm. $e$ is a transcendental number, which was proved by C. Hermite in 1873 for the first time.

$e$ is also defined as the sum of the series

$$\sum _{n=0} ^{\infty} \frac{1}{n!}$$

That means

$$e=\sum _{n=0} ^{\infty} \frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\dots$$

Comments

See also Exponential function; Exponential function, real; Logarithm of a number; Logarithmic function; Transcendental number.

How to Cite This Entry:
E-number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=E-number&oldid=31585
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article