[tex]\it T_4)\\ \\ S=1+\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{10}+\ ...\ +\dfrac{1}{\dfrac{n(n+1)}{2}}\Big|_{\cdot\dfrac{2}{2}}\\ \\ \\ S=\dfrac{2}{2}+\dfrac{2}{2\cdot3}+\dfrac{2}{2\cdot6}+\dfrac{2}{2\cdot10}+\ ...\ +\dfrac{2}{n(n+1)}\\ \\ \\ S=2\Big(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+\dfrac{1}{4\cdot5}+\ ...\ +\dfrac{1}{n(n+1)}\Big)[/tex]
[tex]\it S=2\Big(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\ ...\ +\dfrac{1}{n}-\dfrac{1}{n+1}\Big)\\ \\ \\ S=2\Big(1-\dfrac{1}{n+1}\Big)=2\cdot\dfrac{n+1-1}{n+1}=\dfrac{2n}{n+1}[/tex]
