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 from 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.