純數-數列問題!!!!!!!!!!!!!!純數-數列問題!!

This question is easy.
(a) Consider f'(x)=8/(2x+1)-1
f'(x)>0=>x<7/2
f'(x)<0=>x>7/2
Also f(0)<0 and f(7/2)>0 f(infinity)=-infinity
So f(x)=0 has exactly two real roots
(b) (i) f(alpha)=0 f(beta)=0
So alpha=4ln[(2alpha+1)/2],beta=4ln[(2beta+1)/2]
when n=1, alpha<u_1<beta
when n=k assume that alpha<u_k<beta
u_k+1
=4ln[(2u_k+1)/2]
>4ln[(2alpha+1)/2]
=alpha
Similarly for the other side
So alpha<u_n<beta
(ii) u_n+1-u_n=4ln[(2u_n+1)/2]-u_n>0
[Just consider the graph of f(x)]
Also u_n is bounded by beta, So lim u_n exist
Set the limit is M, then M=4ln[(2M+1)/2]=> M=beta