+ \(n^3=n\cdot n^2>n\left(n^2-1\right)\)
\(\Rightarrow n^3>n\left(n^2+n-n-1\right)\)
\(\Rightarrow n^3>n\left[n\left(n+1\right)-\left(n-1\right)\right]\)
\(\Rightarrow n^3>n\left(n-1\right)\left(n+1\right)\)\(\Rightarrow\frac{1}{n^3}< \frac{1}{\left(n-1\right)n\left(n+1\right)}\)
\(\Rightarrow\frac{1}{n^3}< \frac{1}{2}\left[\frac{\left(n+1\right)-\left(n-1\right)}{\left(n-1\right)n\left(n+1\right)}\right]=\frac{1}{2}\left(\frac{1}{\left(n-1\right)n}-\frac{1}{n\left(n+1\right)}\right)\)
Do đó : \(B< \frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+...+\frac{1}{2018\cdot2019\cdot2020}\)
\(\Rightarrow B< \frac{1}{2}\left(\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+...+\frac{1}{2018\cdot2019}-\frac{1}{2019\cdot2020}\right)\)
\(\Rightarrow B< \frac{1}{2}\left(\frac{1}{2}-\frac{1}{2019\cdot2020}\right)< \frac{1}{4}\)