Web site construction - 604 ANSWERS TO EXERCISES 4.5.3 17. (b) /xl
604 ANSWERS TO EXERCISES 4.5.3 17. (b) /xl -1, 1,22 -2,1,x3 -2,1,. . . , 1, zzn–l -2,1, ~2~ -l/. [Note: One can remove negative parameters from continuants by using the identity Qm+n+l(xl,~~~, xnr -2, ym, . . , Yl) = (-l)m-1Qm+n+2(51,~~~, G-1, Zn -1,1,x -1, -ym,. . . , -y1), from which we obtain Qm+n+l(xlr . . . ,xn, -x,ym, , yl) = -Q m+n+3 (2J1,…,&-1,G-1,1,x -x,1, ym -1, ym-1,. . . ,y1) after a second application.] (c) 1+/1,1,3,1,5,1,… /=1+/2m+l,l/, m>O. 19. The sum for 1 5 k 5 N is log,((l $ z)(N + l)/(N + 1 + z)). 20. Let H = SG, g(z) = (1 + s)G (z), h(z) = (1 + z)H (z). Then (35) implies that h(x + 1)/(x + 2) -Mx)l(x + 1) = -41-t x)r29(1/(1 + x)1/(1 + l/(1 +x,1. 21. p(z) = ~/(~~+1)~+(2-c)/((c-l)z+1)~, Up(z) = l/(z+c) . When c 5 1, the minifnum of ~(z)/U~(s) occurs at z = 0 and is 2c2 5 2. When c > #J = a(&+ l), the minimum occurs at z = 1 and is 5 c$~. When c RZ 1.31266 the values at 2 = 0 and z = 1 are nearly equal and the minimum is > 3.2; the bounds (0.29) (p 5 U p 5 (0.31)n(o are obtained. Still better bounds come from well-chosen linear combinations Tdx) = c %/(X + c3). 23. By the interpolation formula of exercise 4.6.4-15 with 20 = 0, 51 = z, z2 = Z+E, letting t + 0, we have the general identity R ,(z) = (Rn(z) -R,(O))/s + $&(0(s)) for some e(x) between 0 and x, whenever R, is a function with continuous second derivative. Hence in this case R ,(z) = 0(2- ). 24. 00. [A. Khinchin, in Compos. Math. 1 (1935), 361-382, proved that the sum A1 +. . . +A, of the fiist n partial quotients of a real number X will be asymptotically n lg n, for almost all X. Exercise 35 shows that the behavior is different for rational X.1 25. Any union of intervals can be written as a union of disjoint intervals, since we have Uk,lIk =Uk,l(lkUllj