Egyptian fractions

Following the ancient Egyptian number system, rational numbers can be represented by sums of reciprocals, as in 3/7 1/3 + 1/11 + 1/231. With suitable distinct integers a[n] one can represent any number by Sum[1/a[n], {n, ∞}]. The representation is not unique; a[n] = 2^{n}, n (n + 1) and (n + 1)!/n all yield 1. Simple choices for a[n] yield many standard transcendental numbers: n!: - 1; n!^{2}: BesselI[0, 2] - 1; n 2^{n}: Log[2]; n^{2}: π^{2}/6; (3n - 1)(3n - 2): π √3/9; 3 - 16n + 16n^{2}: π/8; n n!: ExpIntegralEi[1] - EulerGamma. (See also page 902.)