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

FormulaNote
Show sourcee=limn(1+1n)ne = \lim_{n\to\infty}\left(1+\tfrac{1}{n}\right)^n-
Show sourcee=limnnn!ne = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}One of so-called Stirling's formulas
Show sourcee=limnn(2πnn!)1/ne = \lim_{n \to \infty} n\cdot\left( \frac{\sqrt{2 \pi n}}{n!} \right)^{1/n}One of so-called Stirling's formulas
Show sourcee=limnn!!ne = \lim_{n\to\infty} \frac{n!}{!n}-
Show sourcee=limn((n+1)n+1nnnn(n1)n1)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

FormulaNote
Show sourcee=2+11+12+23+3e = 2+\frac{1}{1+\frac{1}{2+\frac{2}{3+\frac{3}{\ddots}}}}It' so-called continued fraction.
Show sourcee=n=01n!=10!+11!+12!+13!+14!+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 sourcee=[n=0(1)nn!]1e = \left[ \sum_{n=0}^\infty \frac{(-1)^n}{n!} \right]^{-1}-
Show sourcee=[n=012n(2n)!]1e = \left[ \sum_{n=0}^\infty \frac{1-2n}{(2n)!} \right]^{-1}-
Show sourcee=12n=0n+1n!e = \frac{1}{2} \sum_{n=0}^\infty \frac{n+1}{n!}-
Show sourcee=2n=0n+1(2n+1)!e = 2 \sum_{n=0}^\infty \frac{n+1}{(2n+1)!}-
Show sourcee=n=034n2(2n+1)!e = \sum_{n=0}^\infty \frac{3-4n^2}{(2n+1)!}-
Show sourcee=n=0(3n)2+1(3n)!e = \sum_{n=0}^\infty \frac{(3n)^2+1}{(3n)!}-
Show sourcee=[n=04n+322n+1(2n+1)!]2e = \left[ \sum_{n=0}^\infty \frac{4n+3}{2^{2n+1} (2n+1)!} \right]^2-
Show sourcee=[12π2n=11n2cos(9nπ+n2π29)]1/3e = \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 sourcee=n=1n22(n!)e = \sum_{n=1}^\infty \frac{n^2}{2(n!)}-

Number e as infinite product

FormulaNote
Show sourcee=243685741012141691113158=2n=1i=12n1(2n+2i)i=12n1(2n+2i1)2ne = 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 sourcee=2(21)1/2(2343)1/4(45656787)1/8=2n=1[(2n11)!!]2[(2n)!!]2[(2n1)!!]2[(2n1)!!]22ne = 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}

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=limn(1+1n)ne = \lim_{n\to\infty}\left(1+\tfrac{1}{n}\right)^n
    or as a sum of serie:
    e=n=01n!=10!+11!+12!+13!+14!+e = \sum_{n=0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \dots
  • The approximated value of the number e is:
    e2,718281828459e \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)=exf(x) = exp(x) = e^x

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