From wikipedia:

The number $e$ is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828, and is the limit of $(1~+~1/n)^n$ as $n$ approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series:

$e~=~\displaystyle\sum\limits_{n=0}^{\infty}\dfrac{1}{n!}~=~1+\frac{1}{1}+\frac{1}{1\cdot{2}}+\frac{1}{1\cdot{2}\cdot{3}}+\cdots$

The constant can be defined in many ways. For example, $e$ can be defined as the unique positive number $a$ such that the graph of the function $y=a^x$ has unit slope at $x=0.$ The function $f(x)~=~e^x$ is called the exponential function, and its inverse is the natural logarithm, or logarithm to base $e.$ The natural logarithm of a positive number $k$ can also be defined directly as the area under the curve $y~=~1/x$ between $x=1$ and $x=k,$ in which case, $e$ is the number whose natural logarithm is 1. There are alternative characterizations.

Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, $e$ is not to be confused with $\gamma$ - the Euler-Mascheroni constant, sometimes called simply Euler's constant. The number $e$ is also known as Napier's constant, but Euler's choice of the symbol $e$ is said to have been retained in his honor. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.

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