Let the continued fraction of the first class be reduced to the form

1 1 1

d1 + -- -- -- , then it is convergent if at least one of the

d2 + d3 + d4 + ...

series d3 + d5 + d7 + ..., d2 + d4 + d6 + ... diverges, and oscillates

if both these series converge.

For the convergence of the continued fraction of the second class there

is no complete criterion. The following theorem covers a large number of

important cases.

"If in the infinite continued fraction of the second class

a_n [>=] b_n + 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.