Concrete mathematics : a foundation for computer science

A ANSWERS TO EXERCISES 539
this gives the continued fraction representation of the transcendental number
(b - ‘1 tn3, bP Lnaj, as in exercise 49.
6.82 The sequences of exercise 62 satisfy A-,,, = A,,,, B-~,,, = -B,, and
A,& = Am+, +A,-,;
A,& = Bm+n - ‘L-n ;
BmB, = Am+, -Am-n.
Let fk = Bmk/Amk+l and 9k = Amk/Bmk+,, where 1 = i(n - m). Then
fkt-1 - fk = A~bn/(A2,k+, + AmI and gk ~ gk+l = A~B,/(Azmk+n - A,);
hence we have
2
= ~ - s;,, .
FthLm
6.83 Let p = K(0, al, a2,. . . , a,), so that p/n is the mth convergent to the
continued fraction. Then cx = p/n + (-1 )“‘/nq, where q = K(a, , . , a,,,, 6)
and fi > 1. The points {km} for 0 6 k