# E-number

From Encyclopedia of Mathematics

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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=43591

This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article