设x1=√7,x2=√(7-√7),x3=√(7-√(7+√7)),x4=√(7-√(7+√(7-√7)))…证明{xn

x=√(7-√(7+√(7-√(7+√(7-.
x²=7-√(7+√(7-√(7+√(7-.
x²-7=-√(7+√(7-√(7+√(7-.
7-x²=√(7+√(7-√(7+√(7-.
(7-x²)²=7+√(7-√(7+√(7-.
(7-x²)²-7=√(7-√(7+√(7-.
(7-x²)²-7=x
49-14x²+x^4-7=x
x^4-14x²-x+42=0
x^4+3x^3-3x^3-9x^2-5x²-15x+14x+42=0
(x^4+3x^3)-(3x^3+9x^2)-(5x²+15x)+(14x+42)=0
x^3(x+3)-3x^2(x+3)-5x(x+3)+14(x+3)=0
(x+3)(x^3-3x^2-5x+14)=0
x1=-3
x^3-3x^2-5x+14=0
x^3-2x^2-x^2+2x-7x+14=0
(x^3-2x^2)-(x^2-2x)-(7x-14)=0
x^2(x-2)-x(x-2)-7(x-2)=0
(x-2)(x^2-x-7)=0
x2=2
x^2-x-7=0
x^2-x=7
x^2-x+1/4=7+1/4
(x-1/2)^2=29/4
x-1/2=±√29/2
x=1/2±√29/2
x3=1/2+√29/2
x4=1/2-√29/2
去掉增根,其极限为x2=2