`=>` Tặng bạn
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 :
$\(\dfrac{2}{2^3}< \dfrac{2}{1.2.3}\)$
$\(\dfrac{2}{3^3}< \dfrac{1}{2.3.4}\)$
.................................
$\(\dfrac{2}{n^3}< \dfrac{2}{\left(n-1\right)n\left(n+1\right)}\)$
$\(\Rightarrow2A< \dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+...................+\dfrac{2}{\left(n-1\right)n\left(n+1\right)}\)$
$\(2A< \dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+..............+\dfrac{1}{\left(n-1\right)n}-\dfrac{1}{n\left(n+1\right)}\)$
$\(2A< \dfrac{1}{1.2}-\dfrac{1}{n\left(n+1\right)}\)$
$\(\Rightarrow A< \left(\dfrac{1}{1.2}-\dfrac{1}{n\left(n+1\right)}\right):2\)$
$\(A< \dfrac{1}{4}-\dfrac{1}{2n\left(n+1\right)}\)$
$\(\Rightarrow A< \dfrac{1}{4}\) \(\rightarrowđpcm\)$
