lt Chrystal's _Algebra_ or Serret's _Cours d'Algebre
Superieure_; he may also profitably consult a tract by T. Muir, _The
Expression of a Quadratic Surd as a Continued Fraction_ (Glasgow, 1874).

_The General Continued Fraction._

1. _The Evaluation of Continued Fractions._--The numerators and
denominators of the convergents to the general continued fraction both
satisfy the difference equation u_n = a_{n}u_{n-1} + b_{n}u_{n-2}. When
we can solve this equation we have an expression for the n^{th}
convergent to the fraction, generally in the form of the quotient of two
series, each of n terms. As an example, take the fraction (known as
Brouncker's fraction, after Lord Brouncker)

1 1 squared 3 squared 5 squared 7 squared
-- -- -- -- --
1 + 2 + 2 + 2 + 2 + ...

Here we have

u_{n+1} = 2u_n + (2n-1) squaredu_{n-1},

whence

u_{n+1} - (2n + 1)u_n = -(2n - 1){u_n - (2n - 1)u_{n-1}},

and we readily find that

p_n 1 1 1 1
----- = 1 - -- + -- - -- + ... +- ------,
q_n 3 5 7 2n + 1

whence the value of the fraction taken to infinity is 1/4[pi].

It is always possible to find the value of the n^{th} convergent to a
recurring continued fraction. If r be the number of quotients in the
recurring cycle, we can by writing down the relations connecting the
successive p's and q's obtain a linear relation connecting

p_{nr+m}, p_{(n-1)r+m}, p_{(n-2)r+m},

in which the coefficients are all constants. Or we may proceed as
follows. (We need not consider a fraction with a non-recurring part).
Let the fraction be

a1 a2 a_r a1
-- -- --- --
b1 + b2 + ... + b_r + b1 + ...

p_{nr+m} a1 a2 a_r
Let u_n = --------; then u_n = -- -- ------------, leading
q_{nr+m} b1 + b2 + ... + b_r + u_{n1}

to an equation of the form Au_{n}u_{n-1} + Bu_n + Cu_{n-1} + D = 0,
where A, B, C, D are independent of n, which is readily solved.

2. _The Convergence of Infinite Continued Fractions._--We have seen that
the simple infinite continued fraction converges. The infinite general
continued fraction of the first class cannot diverge for its value lies
between that of its first two convergents. It may, however, oscillate.
We have the relation p_{n}q_{n-1} - p_{n-1}q_n = (-1)^{n}b2b3...b_n,

p_n p_{n-1} b2b3 ... b_n
fr