设X1=1,X2=1+X1/(1+X1),...,Xn=1+Xn-1/(1+Xn-1),
求limXn(请用微积分的方法详细解答)

Xn=1+X(n-1)/[1+X(n-1)], X1=1

1.
显然2≥Xn≥1.


2.用归纳法证明:X(n+1)≥Xn.
ⅰ.X2=1+1/2≥1=X1.

ⅱ.设X(k+1)≥Xk.
X(k+2)-X(k+1)=X(k+1)/[1+X(k+1)]-Xk/[1+Xk]=
=[X(k+1)-Xk]/[(1+Xk)(1+X(k+1)]≥0
所以对于所有n,X(n+1)≥Xn.

3.数列{Xn}递增有上界,所以存在极限.
设Lim{n→∞}Xn=a
X(n+1)=1+Xn/[Xn+1]的两边取极限,
得:a=1+a/(1+a)
==>a^2-a-1 =0,解得 a = (1+√5)/2
==>
lim{n→∞}Xn = (1+√5)/2.