Question

Chững minh rằng:1/5 mũ 2+1/6 mũ 2+1/7 mũ 2+....+1/2007 mũ 2 < 1/4

Answer 1

Đặt \(A=\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}+....+\frac{1}{2007^2}\)\(A< \frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+....+\frac{1}{2006.2007}\)

\(=\frac{5-4}{4.5}+\frac{6-5}{6.5}+....+\frac{2007-2006}{2006.2007}\)

\(=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+.....+\frac{1}{2006}-\frac{1}{2007}\)

\(=\frac{1}{4}-\frac{1}{2007}\)

\(\Leftrightarrow A< \frac{1}{4}-\frac{1}{2007}< \frac{1}{4}\)

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