[tex]\displaystyle \sf S = 1+\frac{7}{10^2}+\frac{7}{10^3}+\frac{7}{10^4}+.. \\\\\\S = 1+\underbrace{\displaystyle \sf \frac{7}{10^2}+\frac{7}{10^3}+\frac{7}{10^4}+..}_{\displaystyle \sf \sum \text{da P.G. infinita}} \\\\\\\\\ \boxed{\begin{array}{I} \sf \text{Soma de uma P.G. infinita } \\\\ \displaystyle \sf S_{\infty } = \frac{a_1}{1-q}\end{array} } \\\\\\ \text{no caso, temos :} \\\\ a_1 = \frac{7}{10^2} \ \ ; \ \ q = \frac{1}{10} \\\\\\ Da{\'i}}:[/tex]

[tex]\displaystyle \sf S = 1+\frac{7}{10^2}+\frac{7}{10^3}+\frac{7}{10^4}+.. \\\\\\ S = 1 + \frac{\displaystyle \sf \frac{7}{10^2}}{\displaystyle 1-\frac{1}{10}} \\\\\\\\ S = 1 + \frac{\displaystyle \frac{7}{10^2}}{\displaystyle \frac{9}{10}} \\\\\\ S = 1+\frac{7}{90} \\\\\\ \huge\boxed{ \sf S = \frac{97}{90} }\checkmark[/tex]

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