“If in the infinite continued fraction of the second class an ≥ bn + 1 for all values of n, it converges to a finite limit not greater than unity.”
3. The Incommensurability of Infinite Continued Fractions.—There is no general test for the incommensurability of the general infinite continued fraction.
Two cases have been given by Legendre as follows:—
If a2, a3, ..., an, b2, b3, ..., bn are all positive integers, then
I. The infinite continued fraction
| b2 | b3 | bn | |||
| a2 | + | a3 | + ... + | an | + ... |
converges to an incommensurable limit if after some finite value of n the condition an ≠ bn is always satisfied.
II. The infinite continued fraction
| b2 | b3 | bn | |||
| a2 | - | a3 | - ... - | an | - ... |
converges to an incommensurable limit if after some finite value of n the condition an ≥ bn + 1 is always satisfied, where the sign > need not always occur but must occur infinitely often.