Question

对数列x1=1\2 x(n+1)=2xn\(1+xn^2) 求xn

Answer 1

X(n+1)=2Xn/[1+(Xn)^2]
令x=2x/(1+x^2)
=>
x=0(略去),x=1，x=-1
=>
X(n+1)+1
2Xn/[1+(Xn)^2]+1
(Xn+1)^2
--------
=
----------------
=
---------------
X(n+1)-1
2Xn/[1+(Xn)^2]-1
-(Xn-1)^2
设An=(Xn+1)/(Xn-1)
=>
A(n+1)=-(An)^2
=>
|A(n+1)|=|-(An)^2|
ln|A(n+1)|=2ln|An|
设ln|An|=Bn
=>
B(n+1)=2Bn
=>
Bn=2^(n-1)*B1
其中B1=ln|A1|=ln|(X1+1)/(X1-1)|=ln3
=>
Bn=ln3*2^(n-1)
=>
|An|
=e^Bn
=e^[ln3*2^(n-1)]
=[e^(ln3)]^[2^(n-1)]
=3^[2^(n-1)]
=>
An
=[(-1)^(n-1)]*3^[2^(n-1)]
=>
(Xn+1)/(Xn-1)=[(-1)^(n-1)]*3^[2^(n-1)]
最后这一步略