e (mathematical Constant) - Representations

Representations

The number e can be represented as a real number in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. The chief among these representations, particularly in introductory calculus courses is the limit

given above, as well as the series

given by evaluating the above power series for ex at x = 1.

Less common is the continued fraction (sequence A003417 in OEIS).

$e =\lim_{n \to \infty} =$

which written out looks like

$2+ \cfrac{1} {1+\cfrac{1} {\mathbf 2 +\cfrac{1} {1+\cfrac{1} {1+\cfrac{1} {\mathbf 4 +\cfrac{1} {1+\cfrac{1} {1+\ddots} } } } } } } = 1+ \cfrac{1} {\mathbf 0 + \cfrac{1} {1 + \cfrac{1} {1 + \cfrac{1} {\mathbf 2 + \cfrac{1} {1 + \cfrac{1} {1 + \cfrac{1} {\mathbf 4 + \cfrac{1} {1 + \cfrac{1} {1 + \ddots} } } } } } } } }$