分类：教育/科学-学习帮助
问题：数列求和问题求证6n/(n+1)(2n+1)&lt;=1+1/4 
求证6n/(n+1)(2n+1)<=1+1/4+1/9+...+1/n^2<5/3

回答：证明：
∵1/n²＜1/(n²-1/4)=4/(4n²-1)=2[1/(2n-1)-1/(2n+1)]
∴∑1/k²＜1+2[1/3-1/5+……+1/(2n-1)-1/(2n+1)]＜1+2/3=5/3
另一方面
1+1/4+1/9+……+1/n²＞1+1/(2×3)+1/(3×4)+……+1/n(n+1)=1-1/(n+1)=n/(n+1)
当n≥3时, n/(n+1)＞6n/(n+1)(2n+1)
当n=1时, 6n/(n+1)(2n+1)=1+1/4+1/9+……+1/n²
当n=2时, 6n/(n+1)(2n+1)＜1+1/4+1/9+……+1/n²
综上, 6n/(n+1)(2n+1)≤1+1/4+1/9+……+1/n²＜5/3