Question

Chứng minh rằng 1/2^3 + 1/3^3 + 1/4^3 + 1/5^3 + ......... + 1/2014^3 < 1/4

Answer 1

Đặt \(A=\frac{1}{2^3}+\frac{1}{3^3}+\frac{1}{4^3}+...+\frac{1}{2014^3}< B=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{2013.2014.2015}\)

Mà \(2B=\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{2013.2014.2015}\)

\(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{2013.2014}-\frac{1}{2014.2015}\)

\(=\frac{1}{2}-\frac{1}{2014.2015}< \frac{1}{2}\)

\(\Rightarrow B< \frac{1}{4}\)

Vậy \(A< \frac{1}{4}\)

Answer 2

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