Nếu \(n>0\Rightarrow\left(n-1\right)n\left(n+1\right)=n^3-n< n^3.\)
\(\Rightarrow VT< \frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{2005.2006.2007}\)
\(\Rightarrow2.VT< \frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{2005.2006.2007}\)
\(\Rightarrow2.VT< \frac{3-1}{1.2.3}+\frac{4-2}{2.3.4}+\frac{5-3}{3.4.5}+...+\frac{2007-2005}{2005.2006.2007}\)
\(\Rightarrow2VT< \frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{2005.2006}-\frac{1}{2006.2007}\)
\(\Rightarrow2.VT< \frac{1}{2}-\frac{1}{2006.2007}\Rightarrow VT< \frac{1}{4}-\frac{1}{2.2006.2007}< \frac{1}{4}\)
