the numerator in the remaining fraction ui decreases and the process of constructing the Engel expansion must terminate in a finite number of steps. Every rational number also has a unique infinite Engel expansion: using the identity the final digit n in a finite Engel expansion can be replaced by an infinite sequence of (n + 1)s without changing its value. For example, This is analogous to the fact that any rational number with a finite decimal representation also has an infinite decimal representation (see 0.999...). An infinite Engel expansion in which all terms are equal is a geometric series. Erdős, Rényi, and Szüsz asked for nontrivial bounds on the length of the finite Engel expansion of a rational number x/y&hairsp;; this question was answered by Erdős and Shallit, who proved that the number of terms in the expansion is O(y1/3 + ε) for any ε > 0. Engel expansions for some well-known constants = (1, 1, 1, 8, 8, 17, 19, 300, 1991, 2492, ...) = (1, 3, 5, 5, 16, 18, 78, 102, 120, 144, ...) = (1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...) And in general, More Engel expansions for constants can be found here. Growth rate of the expansion terms The coefficients ai of the Engel expansion typically exhibit exponential growth; more precisely, for almost all numbers in the interval (0,1], the limit exists and is equal to e. However, the subset of the interval for which this is not the case is still large enough that its Hausdorff dimension is one. The same typical growth rate applies to the terms in expansion generated by the greedy algorithm for Egyptian fractions. However, the set of real numbers in the interval (0,1] whose Engel expansions coincide with their greedy expansions has measure zero, and Hausdorff dimension 1/2. See also Euler's continued fraction formula Notes References . . . . . . External links Mathematical analysis Continued fractions Egyptian fractions