ta có: \(\frac{1}{1.2}>0;\frac{1}{2.3}>0;...;\frac{1}{n.\left(n+1\right)}>0\)
\(\Rightarrow S=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{n.\left(n+1\right)}>0\)
ta có: \(S=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{n.\left(n+1\right)}\)
\(S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{n}-\frac{1}{n+1}\)
\(S=1-\frac{1}{n+1}< 1\)
=> 0 < S < 1
=> S không phải là số tự nhiên