Cn=1/(n^2),Tn为Cn的前n项和,求Tn,并求证Tn

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Cn=1/(n^2),Tn为Cn的前n项和,求Tn,并求证Tn
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Cn=1/(n^2),Tn为Cn的前n项和,求Tn,并求证Tn
Cn=1/(n^2),Tn为Cn的前n项和,求Tn,并求证Tn

Cn=1/(n^2),Tn为Cn的前n项和,求Tn,并求证Tn
主要是放缩,1/1²+1/2²+1/3²+……+1/n²
<1+1/(2²-1)+1/(3²-1)+……+1/n²,用平方差公式展开
=1+1/(1×3)+1/(2×4)+1/(3×5)+……+1/(n-1)(n+1)
=1+1/2×[1-1/3+1/2-1/4+1/3-1/5+……- ……+1/(n-2)-1/n+1/(n-1)-1/(n+1)]
注意,这里是隔项相消.
=1+1/2×(1+1/2-1/n-1/(n+1)
=1+1/2×3/2-1/n-1/(n+1)=7/4-1/n-1/(n+1)<7/4.
证毕.