Mathematical tables: e number formulas
Table shows various methods of calculation or definitions of the so-caller e number.

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# Number e as sequence limit#

 Formula Note Show source$e = \lim_{n\to\infty}\left(1+\tfrac{1}{n}\right)^n$ - Show source$e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}$ One of so-called Stirling's formulas Show source$e = \lim_{n \to \infty} n\cdot\left( \frac{\sqrt{2 \pi n}}{n!} \right)^{1/n}$ One of so-called Stirling's formulas Show source$e = \lim_{n\to\infty} \frac{n!}{!n}$ - Show source$e = \lim_{n\to\infty} \left({\rm }\frac{(n+1)^{n+1}}{n^n} - \frac{n^n}{(n-1)^{n-1}}\right)$ -

# Number e as infinite serie#

 Formula Note Show source$e = 2+\frac{1}{1+\frac{1}{2+\frac{2}{3+\frac{3}{\ddots}}}}$ It' so-called continued fraction. Show source$e = \sum_{n=0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \dots$ - Show source$e = \left[ \sum_{n=0}^\infty \frac{(-1)^n}{n!} \right]^{-1}$ - Show source$e = \left[ \sum_{n=0}^\infty \frac{1-2n}{(2n)!} \right]^{-1}$ - Show source$e = \frac{1}{2} \sum_{n=0}^\infty \frac{n+1}{n!}$ - Show source$e = 2 \sum_{n=0}^\infty \frac{n+1}{(2n+1)!}$ - Show source$e = \sum_{n=0}^\infty \frac{3-4n^2}{(2n+1)!}$ - Show source$e = \sum_{n=0}^\infty \frac{(3n)^2+1}{(3n)!}$ - Show source$e = \left[ \sum_{n=0}^\infty \frac{4n+3}{2^{2n+1} (2n+1)!} \right]^2$ - Show source$e = \left[\frac{-12}{\pi^2} \sum_{n=1}^\infty \frac{1}{n^2} \cos \left( \frac{9}{n\pi+\sqrt{n^2\pi^2-9}} \right) \right]^{-1/3}$ - Show source$e = \sum_{n=1}^\infty \frac{n^2}{2(n!)}$ -

# Number e as infinite product#

 Formula Note Show source$e = 2\cdot\sqrt{\frac{4}{3}}\cdot\sqrt[4]{\frac{6\cdot 8}{5\cdot 7}}\cdot\sqrt[8]{\frac{10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot 13\cdot 15}}\cdots=2\cdot\prod_{n=1}^\infty\sqrt[2^n]{\frac{\prod_{i=1}^{2^{n-1}}(2^n+2i)}{\prod_{i=1}^{2^{n-1}}(2^n+2i-1)}}$ - Show source$e = 2 \left( \frac{2}{1} \right)^{1/2} \left( \frac{2}{3} \frac{4}{3} \right)^{1/4} \left( \frac{4}{5} \frac{6}{5} \frac{6}{7} \frac{8}{7} \right)^{1/8} \cdots =2\prod_{n=1}^\infty\sqrt[2^n]{\frac{[(2^{n-1}-1)!!]^2[(2^n)!!]^2}{[(2^{n-1})!!]^2[(2^n-1)!!]^2}}$ A formula by Nick Pippenger in 1980,$n!!$ is so-called double factorial.

# Some facts#

• The number e is useful in many fields of mathematics, natural sciences and engineering.
• Other common names are Euler's number or Napier's number.
• The number e can be defined in many different ways depending on the context. One of the most common definitions is to present the number e as limit of the sequence:
$e = \lim_{n\to\infty}\left(1+\tfrac{1}{n}\right)^n$
or as a sum of serie:
$e = \sum_{n=0}^\infty \dfrac{1}{n!} = \dfrac{1}{0!} + \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \dfrac{1}{4!} + \dots$
• The approximated value of the number e is:
$e \approx 2,718281828459$
• The e number is the basis of natural logarithm.
• The number e is also related to exponential function:
$f(x) = exp(x) = e^x$

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