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

Beta version#

This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !
⌛ Loading...

Number e as sequence limit#

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#

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#

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 \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:
    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

Tags and links to this website#

What tags this calculator has#


This is permalink. Permalink is the link containing your input data. Just copy it and share your work with friends:

Links to external sites (leaving Calculla?)#

JavaScript failed !
So this is static version of this website.
This website works a lot better in JavaScript enabled browser.
Please enable JavaScript.