The mathematical constant e can be represented in a variety of ways as a real number. Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, e may also be represented as an infinite series, infinite product, or other types of limit of a sequence.
As a continued fraction Euler proved that the number e is represented as the infinite simple continued fraction (sequence A003417 in the OEIS):
e = [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , 1 , 8 , 1 , … , 1 , 2 n , 1 , … ] = 2 + 1 1 + 1 2 + 1 1 + 1 1 + 1 4 + 1 1 + 1 1 + 1 6 + 1 1 + 1 1 + 1 8 +
⋱ {\displaystyle {\begin{aligned}e&=[2;1,2,1,1,4,1,1,6,1,1,8,1,\ldots ,1,2n,1,\ldots ]\\[8pt]&=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{6+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{8+{{} \atop \ddots }}}}}}}}}}}}}}}}}}}}}}}\end{aligned}}}
Here are some infinite generalized continued fraction expansions of e. The second is generated from the first by a simple equivalence transformation.
e = 2 + 1 1 + 1 2 + 2 3 + 3 4 + 4 5 +
⋱ = 2 + 2 2 + 3 3 + 4 4 + 5 5 + 6 6 +
⋱ {\displaystyle e=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {2}{3+{\cfrac {3}{4+{\cfrac {4}{5+{{} \atop \ddots }}}}}}}}}}}=2+{\cfrac {2}{2+{\cfrac {3}{3+{\cfrac {4}{4+{\cfrac {5}{5+{\cfrac {6}{6+{{} \atop \ddots }\,}}}}}}}}}}}
e = 2 + 1 1 + 2 5 + 1 10 + 1 14 + 1 18 +
⋱ = 1 + 2 1 + 1 6 + 1 10 + 1