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$\begin{array}{cll} &\text{Sum}\\ =&\displaystyle \sum^{\infty}_{n=1} \dfrac{7^{n - 1}}{10^n}\\ =&\dfrac{1}{7}\displaystyle\sum^{\infty}_{n = 1} \left(\dfrac{7}{10}\right)^n\\ \stackrel{\text{sum of GS}}{=} &\dfrac{1}{7}\left(\dfrac{\dfrac{7}{10}}{1-\dfrac{7}{10}}\right)\\ =&\dfrac{1}{3} \end{array}$