Cho S=1/2 mũ 2 + 1/3 mũ 2 + 1/4 mũ 2 +…+1/9 mũ 2 Chứng minh rằng: 2/5 04/07/2021 Bởi Kennedy Cho S=1/2 mũ 2 + 1/3 mũ 2 + 1/4 mũ 2 +…+1/9 mũ 2 Chứng minh rằng: 2/5 { "@context": "https://schema.org", "@type": "QAPage", "mainEntity": { "@type": "Question", "name": " Cho S=1/2 mũ 2 + 1/3 mũ 2 + 1/4 mũ 2 +...+1/9 mũ 2 Chứng minh rằng: 2/5
$S=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+…+\frac{1}{9^2}\\= \frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+…+\frac{1}{9.9}\\S<\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+…+\frac{1}{8.9}=S_1\\ S_1=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+…+\frac{1}{8.9}\\ =\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+…+\frac{1}{8}-\frac{1}{9}\\ =1-\frac{1}{9}\\ =\frac{8}{9}(1)\\ S>\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+…+\frac{1}{9.10}=S_2\\ S_2=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+…+\frac{1}{9.10}\\ =\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+…+\frac{1}{9}-\frac{1}{10}\\ =\frac{1}{2}-\frac{1}{10}\\ =\frac{2}{5}(2)\\ (1)(2)=>\frac{2}{5}<S<\frac{8}{9}$ Bình luận
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$S=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+…+\frac{1}{9^2}\\= \frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+…+\frac{1}{9.9}\\S<\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+…+\frac{1}{8.9}=S_1\\ S_1=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+…+\frac{1}{8.9}\\ =\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+…+\frac{1}{8}-\frac{1}{9}\\ =1-\frac{1}{9}\\ =\frac{8}{9}(1)\\ S>\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+…+\frac{1}{9.10}=S_2\\ S_2=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+…+\frac{1}{9.10}\\ =\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+…+\frac{1}{9}-\frac{1}{10}\\ =\frac{1}{2}-\frac{1}{10}\\ =\frac{2}{5}(2)\\ (1)(2)=>\frac{2}{5}<S<\frac{8}{9}$