om which --- - ------- = (-1)^n ------------, and the limit of the
q_n q_{n-1} q_{n}q_{n-1}

right-hand side is not necessarily zero.

The tests for convergency are as follows:

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.

Two cases have been given by Legendre as follows:--

If a2, a3, ..., a_n, b2, b3, ...,b_n are all positive integers, then

b2 b3 b_{n}
I. The infinite continued fraction -- -- ----- converges
a2 + a3 + ... + a_{n} + ...

to an incommensurable limit if after some finite value of n the condition
a_{n} [not <] b_{n} is always satisfied.

b2 b3 b_{n}
II. The infinite continued fraction -- -- -----
a2 - a3 - ... - a_{n} - ...

converges to an incommensurable limit if after some finite value of n
the condition a_{n} [>=] b_{n} + 1 is always satisfied, where the sign >
need not always occur but must occur _infinitely often_.

_Continuants._

The functions p_{n} and q_{n}, regarded as functions of a1, ..., a_{n},
b2, ..., b_{n} determined by the relations

p_{n} = a_{n}p_{n-1} + b_{n}p_{n-2},
q_{n} = a_{n}q_{n-1} + b_{n}q_{n-2},

with the conditions p1 = a1, p0 = 1; q2 = a2, q1 = 1, q0 = 0, have been
studied under the name of _continuants_. The notation adopted is

/ b2,...,b_{n}\
p_{n} = K ( ),
\a1, a2,...,a_{n}/

and it is evident that we have

/ b3,...,b_{n}\
q_{n} = K ( ).
\a2, a3,...,a_{n}/

The theory of continuants is due in the first place