若函数f(x)=X/(1+X),且f(n)(x)=f(f(...f(x) 有n个,则f(2011)(

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f(1)(1) = f(1) = 1/2;
f(2)(1) = f(f(1)(1)) = 1/3;
f(3)(1) = f( f(2)(1) ) = f( f(f(1)(1)) ) = 1/4;
.f(n-1)(1) = f( f(n-2)(1) ) = 1/n;
f(n)(1) = f( f(n-1)(1) ) = 1/(n+1);
上式是f(n)(1)的通项公式,用数学归纳法可以证明.
所以f(2011)(1) = 1/2012;