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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=lim⁡n→∞(1+1n)ne = \lim_{n\to\infty}\left(1+\tfrac{1}{n}\right)^n-
Show sourcee=lim⁡n→∞nn!ne = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}One of so-called Stirling's formulas
Show sourcee=lim⁡n→∞n⋅(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=lim⁡n→∞n!!ne = \lim_{n\to\infty} \frac{n!}{!n}-
Show sourcee=lim⁡n→∞((n+1)n+1nn−nn(n−1)n−1)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+3⋱e = 2+\frac{1}{1+\frac{1}{2+\frac{2}{3+\frac{3}{\ddots}}}}It' so-called continued fraction.
Show sourcee=∑n=0∞1n!=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=0∞1−2n(2n)!]−1e = \left[ \sum_{n=0}^\infty \frac{1-2n}{(2n)!} \right]^{-1}-
Show sourcee=12∑n=0∞n+1n!e = \frac{1}{2} \sum_{n=0}^\infty \frac{n+1}{n!}-
Show sourcee=2∑n=0∞n+1(2n+1)!e = 2 \sum_{n=0}^\infty \frac{n+1}{(2n+1)!}-
Show sourcee=∑n=0∞3−4n2(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=0∞4n+322n+1(2n+1)!]2e = \left[ \sum_{n=0}^\infty \frac{4n+3}{2^{2n+1} (2n+1)!} \right]^2-
Show sourcee=[−12π2∑n=1∞1n2cos⁡(9nπ+n2π2−9)]−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=1∞n22(n!)e = \sum_{n=1}^\infty \frac{n^2}{2(n!)}-

Number e as infinite product#

FormulaNote
Show sourcee=2⋅43⋅6⋅85⋅74⋅10⋅12⋅14⋅169⋅11⋅13⋅158⋯=2⋅∏n=1∞∏i=12n−1(2n+2i)∏i=12n−1(2n+2i−1)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⋯=2∏n=1∞[(2n−1−1)!!]2[(2n)!!]2[(2n−1)!!]2[(2n−1)!!]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=lim⁡n→∞(1+1n)ne = \lim_{n\to\infty}\left(1+\tfrac{1}{n}\right)^n
    or as a sum of serie:
    e=∑n=0∞1n!=10!+11!+12!+13!+14!+…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≈2,718281828459e \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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