Given:
The series is 1/3−2/21+4/147−8/1029+··
Explanation:
For the given series, the first term is,
[tex]a=\frac{1}{3}[/tex]The common ratio is,
[tex]\begin{gathered} r=\frac{-\frac{2}{21}}{\frac{1}{3}} \\ =-\frac{2}{21}\cdot\frac{3}{1} \\ =-\frac{2}{7} \end{gathered}[/tex]The formula for the sum of infinite series is,
[tex]S_{\infty}=\frac{a}{1-r}[/tex]Substitute the values in the formula to determine the sum of infinite series.
[tex]\begin{gathered} S_{\infty}=\frac{\frac{1}{3}}{1-(-\frac{2}{7})} \\ =\frac{\frac{1}{3}}{\frac{9}{7}} \\ =\frac{1}{3}\times\frac{7}{9} \\ =\frac{7}{27} \end{gathered}[/tex]Answer: 7/27
