\(A=\dfrac{1}{2^3}+\dfrac{1}{3^3}+...+\dfrac{1}{n^3}+\dfrac{1}{2017^3}\)
\(A=\dfrac{1}{8}+\dfrac{1}{3^3}+...+\dfrac{1}{n^3}+\dfrac{1}{2017^3}>\dfrac{1}{8}>\dfrac{1}{12}\left(1\right)\)
Xét thừa số tổng quát: \(\dfrac{1}{n^3}< \dfrac{1}{n^3-n}=\dfrac{1}{n\left(n^2-1\right)}=\dfrac{1}{\left(n-1\right)n\left(n+1\right)}\)
Hay:
\(A< \dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+...+\dfrac{1}{\left(n-1\right)n\left(n+1\right)}+...+\dfrac{1}{2016.2017.2018}\)
\(A< \dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+..+\dfrac{1}{\left(n-1\right)n}-\dfrac{1}{n\left(n+1\right)}+...+\dfrac{1}{2016.2017}-\dfrac{1}{2017.2018}\right)\)
\(A< \dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{2017.2018}\right)=\dfrac{1}{4}-\dfrac{1}{2.2017.2018}< \dfrac{1}{4}< \dfrac{505}{5028}\left(2\right)\)
Từ (1) và (2) ta có đpcm