已知数列{an}满足a1=4/3,且an+1=〔4(n+1)an〕/(3an+n) (n∈N*)已知数列{an}满足a1=4/3,且an+1=〔4(n+1)an〕/(3an+n)(n∈N*).(1)求1/a1+2/a2+…+n/an的值;(2)求证:a1+a2/2+a3/3+…+an/n≤ n+ 7/12-(1/4)^n
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已知数列{an}满足a1=4/3,且an+1=〔4(n+1)an〕/(3an+n) (n∈N*)
已知数列{an}满足a1=4/3,
且an+1=〔4(n+1)an〕/(3an+n)
(n∈N*).
(1)求1/a1+2/a2+…+n/an的值;
(2)求证:a1+a2/2+a3/3+…+an/n
≤ n+ 7/12-(1/4)^n
1,两边同时取倒数,得1/a(n+1)=3/[4(n+1)]+n/[4(n+1)an],两边同乘以(n+1),得
(n+1)/a(n+1)=n/(4an)+3/4,所以(n+1)/a(n+1)-1=(1/4)[n/an-1],设bn=(n/an)-1,所以
b(n+1)=(n+1)/a(n+1)-1,所以b(n+1)=(1/4)bn,所以bn是以1/4为公比的等比数列.
求得bn=-(1/4)^n,所以n/an=1-(1/4)^n.
2,放缩法,把an/n放缩成一个可求和的数列.
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