Question

设﹛Xn﹜满足-1＜X0＜0,Xn+1=Xn∧2+2Xn（n=0,1,2,…）,证明﹛Xn﹜收敛,并求极限

Answer 1

Xn+1=Xn∧2+2Xn=(xn+1)^2-1>=-1xn有下界-1由于Xn+1=Xn∧2+2Xnxn+1-xn=xn^2+xn=xn(xn+1)所以Xn=Xn-1∧2+2Xn-1利用饥陆兆数学归纳x1=x0^2+2x0=(x0+1)^2-11. x1-x0=x0^2+x0=x0(x0+1)-1即悉吵xn+xn-1+2>0所以烂租Xn+1-xn=(xn-xn-1)(xn...