[tex]\it \dfrac{1}{k}-\dfrac{1}{k+2} = \dfrac{k+2-k}{k(k+2)} = \dfrac{2}{k(k+2)} \Longrightarrow \dfrac{1}{k(k+2)} = \dfrac{1}{2}\left( \dfrac{1}{k}-\dfrac{1}{k+2}\right)
\\ \\ \\
S = \dfrac{1}{2}\left(1- \dfrac{1}{3} + \dfrac{1}{3} -\dfrac{1}{5} +\ ...\ +\dfrac{1}{89}-\dfrac{1}{91}\right) =\dfrac{1}{2}\left(1-\dfrac{1}{91}\right) =
\\ \\ \\
= \dfrac{1}{2}\cdot\dfrac{90}{91} = \dfrac{45}{91}[/tex]
