Question

For two fixed positive numbers a, b, 0 < a < b (for example, a = 1, b = 2), let a = V ab and b = - - Define aj+1 = a;b; and b;+1 = - T for any positive interger j. a+b (1) Prove that the sequence {aj} is convergent. (2) Prove that the sequence {b;} is convergent. (3) Prove that lim a;= lim bj. 1700

Please explain in full detail!

Answer 1

2 Given data a, b, olaLb 9 = Vab and bia ato ajti = Vaj b; and b; t1 = a; thi By mean inequality atb z jab biza, • ajti = vaj bj 3 ajt bi biti bjt ajti [using (a)) b; z aj tas aj ti = raj b; z vaj b; ajti z rajz - aj ajti zaj (b) • ajti z ajt bi < bi+b; biti sbi (6) combining (a), (b) & (C) [uréng ca)) We get, bebiz be bzb, z bz 2...-.bý> bjt, zajt, Za; z...20, Zad) C) Erom (6) and (d) sequence {by} is monotone increasing sequence and bounded above by b - By Monotone convergence theorem {bi} is convergent.. (2) From (c) and (d) sequence { brj} is decreasing sequence and bounded below by a :. By monotone convergence the grem {bj} ý convergent, (3) : {aj } ý convergent 7 alfinite) I..|- lim aj za (e) ajti = raj b; 2) ajti ? = a; b; bj = aj + 1² lingen by a malinama aja i a [wing les] lim b;=a 17 nyo ou a lim bj = tim a;