It easily follows that the continued fraction is incommensurable if a/b, c/d, etc., being at first greater than unity, become and continue less than unity after some one point. Say that i/k, l/m,... are all less than unity. Then the fraction i/k+ l/m+ ... is incommensurable, as proved: let it be κ. Then g/(h + κ) is incommensurable, say λ; e/(f + λ) is the same, say μ; also c/(d + μ), say ν, and a/(b + ν), say ρ. But ρ is the fraction a/b+ c/d+ ... itself; which is therefore incommensurable.
Let φz represent
| 1 + | a z | + | a2 2z(z+1) | + | a3 2·3·z(z+1)(z+2) | + ... |
Let z be positive: this series is convergent for all values of a, and approaches without limit to unity as z increases without limit. Change z into z + 1, and form φz - φ(z+1): the following equation will result—
| φz - φ(z+1) = | a z(z+1) | φ(z+2) |
| or a = | a z | φ(z+1) φz | · z + | a z | φ(z+1) φz | · | a z+1 | φ(z+2) φ(z+1) |
| or a = ψz | z + ψ(z+1) |
ψz being (a/z)(φ(z+1)/φz); of which observe that it diminishes without limit as z increases without limit. Accordingly, we have
| ψz = | a z+ | ψ(z+1) = | a z+ | a (z+1)+ | ψ(z+2) = | a z+ | a (z+1)+ | a (z+2)+ | ψ(z+3) |