Postgraduate entrance examination mathematics, sequence limit, how to do the second question? process

(1) let f (x) = x n+x (n-1)+..........+x-1,

Then f' (x) = NX (n-1)+(n-1) x (n-2)++3x2+2x+1,

Obviously, f '(x) is always positive at (0, 1), so the function f(x) is strictly monotonically increasing at (0, 1).

f( 1/2)=( 1/2)n+( 1/2)(n- 1)+( 1/2)-65438+。 0 ,

f( 1)= 1+ 1+......+ 1- 1 = n- 1 & gt； 0 ,

So f(x) has a unique real root in the interval (1/2, 1).

(2) multiply both sides of the equation by x- 1 to get X (n+ 1)-X = X- 1, so X (n+ 1) = 2x- 1,

So xn (n+ 1) = 2xn- 1,

According to (1),1/2 < xn <; 1, so the limit of xn (n+ 1) is 0 when n tends to infinity.

Taking the limit where n on both sides of the above formula tends to infinity, it can be concluded that xn limit exists, which is 1/2.