Question

[Trần Thùy Linh]

CMR : A = 1/2^3 + 1/3^3 + .................... + 1/n^3 <1/4

Answer 1

[Nguyễn Thanh Hằng]

Ta có :

\(A=\dfrac{1}{2^3}+\dfrac{1}{3^3}+\dfrac{1}{4^3}+......................+\dfrac{1}{n^3}\)

\(2A=\dfrac{2}{2^3}+\dfrac{2}{3^3}+\dfrac{2}{4^3}+.....................+\dfrac{2}{n^3}\)

Vì :

\(\dfrac{2}{2^3}< \dfrac{2}{1.2.3}\)

\(\dfrac{2}{3^3}< \dfrac{1}{2.3.4}\)

.................................

\(\dfrac{2}{n^3}< \dfrac{2}{\left(n-1\right)n\left(n+1\right)}\)

\(\Rightarrow2A< \dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+...................+\dfrac{2}{\left(n-1\right)n\left(n+1\right)}\)

\(2A< \dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+..............+\dfrac{1}{\left(n-1\right)n}-\dfrac{1}{n\left(n+1\right)}\)

\(2A< \dfrac{1}{1.2}-\dfrac{1}{n\left(n+1\right)}\)

\(\Rightarrow A< \left(\dfrac{1}{1.2}-\dfrac{1}{n\left(n+1\right)}\right):2\)

\(A< \dfrac{1}{4}-\dfrac{1}{2n\left(n+1\right)}\)

\(\Rightarrow A< \dfrac{1}{4}\) \(\rightarrowđpcm\)

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