高中数列求和中不等式恒成立问题

问题描述:

高中数列求和中不等式恒成立问题
bn=1/n(n+2),bn前n项和为Tn,求证Tn

数学人气:372 ℃时间:2020-08-30 06:35:40
优质解答
Tn=1/(1*3)+1/(2*4)+1/(3*5)+……+1/[(n-1)(n+1)]+1/n(n+2)
=1/2[1-1/3+1/2-1/4+1/3-1/5+……+1/(n-1)-1/(n+1)+1/n-1/(n+2)]
=1/2[1+1/2-1/(n+1)-1/(n+2)]
=3/4-1/(2n+2)-1/(2n+4)
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Tn=1/(1*3)+1/(2*4)+1/(3*5)+……+1/[(n-1)(n+1)]+1/n(n+2)
=1/2[1-1/3+1/2-1/4+1/3-1/5+……+1/(n-1)-1/(n+1)+1/n-1/(n+2)]
=1/2[1+1/2-1/(n+1)-1/(n+2)]
=3/4-1/(2n+2)-1/(2n+4)