Question Number 103260 by 675480065 last updated on 13/Jul/20

Answered by Rio Michael last updated on 13/Jul/20

  { ((u_0  = 1)),((u_(n+1)  = ((2u_n v_n )/(u_n + v_n )))) :} and  { ((v_0  = 2)),((v_(n+1 ) = ((u_n  + v_n )/2))) :}  ∀ n ∈ N  (a) n ∈ N ⇒ 2u_n v_n  > 0 andu_n  + v_n  > 0 since u_n  and v_n  > 0  hence u_n  and v_n  are strickly positive sequences.  (b) w_n  = v_n −u_n  ⇒ w_(n+1)  = v_(n+1) −u_(n+1)   ⇒ w_(n+1)  = ((u_n  + v_n )/2)−((2u_n v_n )/(u_n + v_n )) = ((u_n ^2  + 2u_n v_n  + v_n ^2  −4u_n v_n )/(2(u_n  + v_n ))) = ((u_n ^2   −2u_n v_n + v_n ^2 )/(2(u_n  + v_n )))  ⇒ w_(n+1)  = (((u_n −v_n )^2 )/(2(u_(n ) + v_n ))) ⇒ w_(n+1)  ≤ (1/2)(v_n −u_n ) = (1/2)w_n   hence 0 ≤ w_(n+1) ≤ (1/2)w_n   (c)lim_(n→∞) w_n  = 0