Definition:Euler's Number
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Definition
As the Limit of a Sequence
The sequence $\left \langle {x_n} \right \rangle$ defined as $x_n = \left({1 + \dfrac 1 n}\right)^n$ converges to a limit.
That limit is called Euler's Number and is denoted $e$.
Its value is approximately:
- $2.71828 \ 18284 \ 59045 \ 23536 \ 0287 \ldots$
This sequence is A001113 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
As the Limit of a Series
The series $\displaystyle \sum_{n=0}^\infty \frac 1 {n!}$ also converges to the same limit.
As the Base of the Natural Logarithm
The number $e$ can also be defined as the number satisfied by $\ln e = 1$.
In Terms of the Exponential Function
We can also define $e$ as:
- $e := \exp 1 = e^1$
Equivalence of Definitions
It is shown in Equivalence of Definitions of Euler's Number that all the methods given here for specifying $e$ are logically equivalent.
Comment
This is the most famous irrational constant in mathematics after $\pi$, and equally far-reaching in scope and usefulness.
The proof that it is irrational is straightforward.
Source of Name
This entry was named for Leonhard Paul Euler.
