Question

证明数列收敛,并求极限设a > 0 ,0 < X1< 1/a ,X n+1= X n (2 - a * X n) (n=

Answer 1

Xn+1=Xn×(2-a*Xn)=-a×(Xn-1/a)²+1/a

→ (1/a-Xn+1)=a×(1/a-Xn)²

令Yn=1/a-Xn,则Yn+1=a×Yn² (Y1=1/a-X1,n≥2)

∴Yn+1=a^(2*n-1)×Y1^(2*n)=1/a×(a*Y1)^(2*n)

∴Xn+1=1/a-1/a×(a*Y1)^(2*n)

∵Y1=1/a-X1,即,0＜Y1＜1/a

∴0＜a*Y1＜1

∴0＜(a*Y1)^(2*n)＜1

∴0＜Xn+1＜1/a

当n→+∞时,(a*Y1)^(2*n)→0,Xn+1→1/a