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 :
\(\dfrac{2}{2^3}< \dfrac{2}{1.2.3}\)
\(\dfrac{2}{3^3}< \dfrac{1}{2.3.4}\)
.................................
\(\dfrac{2}{n^3}< \dfrac{2}{\left(n-1\right)n\left(n+1\right)}\)
\(\Rightarrow2A< \dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+...................+\dfrac{2}{\left(n-1\right)n\left(n+1\right)}\)
\(2A< \dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+..............+\dfrac{1}{\left(n-1\right)n}-\dfrac{1}{n\left(n+1\right)}\)
\(2A< \dfrac{1}{1.2}-\dfrac{1}{n\left(n+1\right)}\)
\(\Rightarrow A< \left(\dfrac{1}{1.2}-\dfrac{1}{n\left(n+1\right)}\right):2\)
\(A< \dfrac{1}{4}-\dfrac{1}{2n\left(n+1\right)}\)
\(\Rightarrow A< \dfrac{1}{4}\) \(\rightarrowđpcm\)
~ Chúc bn học tốt ~
