358 ARITHMETIC 4.5.3 2. [M.8] Evaluate the matrix (Affordable web hosting)
358 ARITHMETIC 4.5.3 2. [M.8] Evaluate the matrix product 3. [A&Y] What is the value of 4. [A420] Prove Eq. (8). 5. [HM25] Let xi, x2, . . . be a sequence of real numbers that are each greater than some positive real number E. Prove that the infinite continued fraction /xl, 22,. . . / = lim,,, /xl,. . . , xn/ exists. Show also that 1×1, x2,. . . / need not exist if we assume only that x3 > 0 for all j. 6. [MZS ] Prove that the regular continued fraction expansion of a number is unique in the following sense: If Bi, B2, . . . are positive integers, then the infinite continued fraction /BI, B2, . . . / is an irrational number X between 0 and 1 whose regular con- tinued fraction has A, = B, for all n 2 1; and if BI, . . . , B, are positive integers with B, > 1, then the regular continued fraction for X = /B1,. . . , Bm/ has A, = B, for 1 5 n 2 m. 7. [M96] Find all permutations p(l)p(2). . . p(n) of the integers {1,2,. . . , n} such that Qn(xl, x2,. . , xn) = Qn(xp(l), zp(2), . , x~(~)) holds for all XI, x2, . . . , xn. 8. [.W?O] Show that -l/Xn = /An,. . . , AI, -Xl, whenever X, is defined, in the regular continued fraction process. 9. [MZZ] Show that continued fractions satisfy the following identities: a) /xl,. . . , xn/ = /xl,. . . ,Xk + /Xk+l,. . . , ?A//, l