Engel expansion

The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers ${\displaystyle (a_{1},a_{2},a_{3},\dots )}$ such that

${\displaystyle x={\frac {1}{a_{1}}}+{\frac {1}{a_{1}a_{2}}}+{\frac {1}{a_{1}a_{2}a_{3}}}+\cdots ={\frac {1}{a_{1}}}\!\left(1+{\frac {1}{a_{2}}}\!\left(1+{\frac {1}{a_{3}}}\left(1+\cdots \right)\right)\right)}$

For instance, Euler's number e has the Engel expansion^[1]

1, 1, 2, 3, 4, 5, 6, 7, 8, ...

corresponding to the infinite series

${\displaystyle e={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+{\frac {1}{1\cdot 2\cdot 3\cdot 4}}+\cdots }$

Rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion. If x is rational, its Engel expansion provides a representation of x as an Egyptian fraction. Engel expansions are named after Friedrich Engel, who studied them in 1913.

An expansion analogous to an Engel expansion, in which alternating terms are negative, is called a Pierce expansion.