Sylvester's Sequence - Connection With Egyptian Fractions

Connection With Egyptian Fractions

The unit fractions formed by the reciprocals of the values in Sylvester's sequence generate an infinite series:

The partial sums of this series have a simple form,

This may be proved by induction, or more directly by noting that the recursion implies that

so the sum telescopes

Since this sequence of partial sums (s_j-2)/(s_j-1) converges to one, the overall series forms an infinite Egyptian fraction representation of the number one:

One can find finite Egyptian fraction representations of one, of any length, by truncating this series and subtracting one from the last denominator:

The sum of the first k terms of the infinite series provides the closest possible underestimate of 1 by any k-term Egyptian fraction. For example, the first four terms add to 1805/1806, and therefore any Egyptian fraction for a number in the open interval (1805/1806,1) requires at least five terms.

It is possible to interpret the Sylvester sequence as the result of a greedy algorithm for Egyptian fractions, that at each step chooses the smallest possible denominator that makes the partial sum of the series be less than one. Alternatively, the terms of the sequence after the first can be viewed as the denominators of the odd greedy expansion of 1/2.