已知数列{an}的前n项和为Sn=n^2+1,数列{bn}满足:bn=2/(an+1),且前n项和为Tn,设Cn=T(2n+1)-Tn.若对n>=k时,总有Cn

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已知数列{an}的前n项和为Sn=n^2+1,数列{bn}满足:bn=2/(an+1),且前n项和为Tn,设Cn=T(2n+1)-Tn.若对n>=k时,总有Cn
已知数列{an}的前n项和为Sn=n^2+1,数列{bn}满足:bn=2/(an+1),且前n项和为Tn,设Cn=T(2n+1)-Tn.
若对n>=k时,总有Cn

已知数列{an}的前n项和为Sn=n^2+1,数列{bn}满足:bn=2/(an+1),且前n项和为Tn,设Cn=T(2n+1)-Tn.若对n>=k时,总有Cn
Sn=n^2+1
Sn-1=(n-1)^2+1
∴an=2n-1
bn=2/(2n-1+1)=1/n
Cn=bn+1+...+b2n+1=1/(n+1)+1/(n+2)..+1/(2n+1)
Cn+1=1/(n+2)+...+1/(2n+1)+1/(2n+2)+1/(2n+3)
Cn-Cn+1=1/(n+1)-1/(2n+2)-1/(2n+3)=1/(2n+2)-1/(2n+2)+1/(2n+2)-1/(2n+3)=1/(2n+2)(2n+3)>0
∴Cn为递减函数
C2=38/60>16/21>C3=319/420
∴k=3

(1)a1=2,an=Sn-Sn-1=2n-1(n≥2).
∴bn= 1 n (n≥2) 2 3 (n=1) (2)∵cn=bn+1+bn+2+…+b2n+1
=1 n+1 +1 n+2 +…+1 2n+1 ,
∴cn+1-cn=1 2n+2 +1 2n+3 -1 n+1 <0,