[tex]\Large\boxed{\begin{array}{l}\underline{\rm Frac_{\!\!,}\tilde ao\,geratriz\,pelo\,m\acute etodo}\\\underline{\rm da\,soma\,dos\,termos\,de\,uma\,PG\,infinita.}\\\sf 0,555...=\underbrace{\sf0,5...+0,05....+0,005...}_{soma\,dos\,termos\,da\,PG\,infinita.}\\\\\sf a_1=0,5=\dfrac{5}{10}\\\\\sf a_2=0,05=\dfrac{5}{100}\\\\\sf q=\dfrac{a_2}{a_1}=\dfrac{\frac{5}{10\backslash\!\!\!0}}{\frac{5}{1\backslash\!\!\!0}}\\\\\sf q=\dfrac{\backslash\!\!\!5}{10}\cdot\dfrac{1}{\backslash\!\!\!5}=\dfrac{1}{10}\end{array}}[/tex]
[tex]\Large\boxed{\begin{array}{l}\sf S_n=\dfrac{a_1}{1-q}\\\\\sf S_n=\dfrac{\frac{5}{10}}{1-\frac{1}{10}}\\\\\sf S_n=\dfrac{\frac{5}{\red{\diagdown\!\!\!\!\!\!}10}}{\frac{9}{\red{\diagdown\!\!\!\!\!\!}10}}\\\\\sf S_n=\dfrac{5}{9}\end{array}}[/tex]
[tex]\Large\boxed{\begin{array}{l}\underline{\rm Frac_{\!\!,}\tilde ao\,geratriz\,pelo\,m\acute etodo}\\\underline{\rm das\,equac_{\!\!,}\tilde oes}\\\sf x=0,555...\cdot10\\\sf 10x=5,555...\\-\underline{\begin{cases}\sf 10x=5,555...\\\sf x=0,555....\end{cases}}\\\sf 9x=5\\\\\sf x=\dfrac{5}{9}\end{array}}[/tex]
