题目

放缩法证不等式
求证：3/2-1/(n+1)＜1+1/(2^2)+1/(3^2)+……+1/n^2＜2-1/n

提问时间：2021-03-03

答案

1+1/2²+1/3²+...+1/n²
>1+1/(2×3)+1/(3×4)+...+1/[n(n+1)]
=1+(1/2-1/3)+(1/3-1/4)+...+(1/n-1/(n+1))
=1+1/2-1/3+1/3-1/4+...+1/n-1/(n+1)
=(3/2)-1/(n+1)
1+1/2²+1/3²+...+1/n²
<1+1/(1×2)+1/(2×3)+...+1/[(n-1)n]
=1+(1-1/2)+(1/2-1/3+...+(1/(n-1)-/n)
=1++1-1/2+1/2-1/3+...+1/(n-1)-1/n
=2-1/n
综合得,(3/2)-1/(n+1)＜1+1/(2^2)+1/(3^2)+……+1/n^2＜2-1/n