Difference between revisions of "E-number"
From Encyclopedia of Mathematics
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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: | ||
| − | $$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n=2.718281828459045\ | + | $$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n=2.718281828459045\dots;$$ |
| − | it is the base for the natural logarithm. $e$ is a transcendental number, which was proved by C. Hermite in 1873 for the first time | + | it is the base for the natural logarithm. $e$ is a transcendental number, which was proved by C. Hermite in 1873 for the first time. |
| + | $e$ is also defined as the sum of the series | ||
| + | $$\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$$ | ||
====Comments==== | ====Comments==== | ||
See also [[Exponential function|Exponential function]]; [[Exponential function, real|Exponential function, real]]; [[Logarithm of a number|Logarithm of a number]]; [[Logarithmic function|Logarithmic function]]; [[Transcendental number|Transcendental number]]. | See also [[Exponential function|Exponential function]]; [[Exponential function, real|Exponential function, real]]; [[Logarithm of a number|Logarithm of a number]]; [[Logarithmic function|Logarithmic function]]; [[Transcendental number|Transcendental number]]. | ||
Latest revision as of 15:10, 30 December 2018
The limit of the expression $(1+1/n)^n$ as $n$ tends to infinity:
$$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
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