# Euler's number

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Euler's number (also known as Napier's constant), \(e\), is a mathematical constant, which is approximately equal to

\[ 2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713829178...\]

It can be expressed as the limit, \( \displaystyle e = \lim_{n\to\infty} \left( 1 + \dfrac{1}{n} \right)^n \), and it can also be expressed as the infinite sum \( \displaystyle \sum_{j=0}^\infty \dfrac{1}{j!} = \dfrac{1}{0!} + \dfrac{1}{1!} + \dfrac{1}{2!} + \cdots \).

It is not be confused with \(\gamma\) (Euler-Mascheroni constant).

## Proof of Equivalence of Definitions

\(\displaystyle\lim_{n\to\infty} \left(1+\frac 1n\right)^n=\sum_{j=0}^\infty \frac 1{j!}\)

Apply binomial expansion on the left hand side, we have:

\(\displaystyle\;\;\;\;\lim_{n\to\infty}(1+\frac 1n)^n\\ \displaystyle=\lim_{n\to\infty}(1+\frac n{1!}\cdot\frac 1n+\frac{n(n-1)}{2!} \frac 1{n^2}+\frac{n(n-1)(n-2)}{3!}\frac 1{n^3}+\cdots)\\ \displaystyle=\lim_{n\to\infty}(1+\frac 1{1!}\frac nn+\frac 1{2!}\frac{n(n-1)}{n^2}+\frac 1{3!}\frac{n(n-1)(n-2)}{n^3}+\cdots)\\ \displaystyle=\lim_{n\to\infty}(1+\frac 1{1!}(1)+\frac 1{2!}(1)(1-\frac 1n)+\frac 1{3!}(1)(1-\frac 1n)(1-\frac 2n)+\cdots)\\ \displaystyle=1+\frac 1{1!}+\frac 1{2!}+\frac 1{3!}+\cdots\\ \displaystyle=\frac 1{0!}+\frac 1{1!}+\frac 1{2!}+\frac 1{3!}+\cdots\\ \displaystyle=\sum_{j=0}^\infty \frac 1{j!}\)

## Applications

Complex Numbers: \[e^{i\theta} = \cos\theta + i\sin \theta\]