Find the sum of the infinite series. $\sum _ { j = 1 } ^ { \infty } \left( \frac { 7 } { 10 } \right) ^ { \rangle }$
A) $- \frac { 7 } { 3 } j$
B) $\frac { 7 } { 3 }$
C) $- \frac { 7 } { 10 } j$
D) $\frac { 7 } { 10 }$
E) $- \frac { 7 } { 3 }$

Question

Find the sum of the infinite series.  $\sum _ { j = 1 } ^ { \infty } \left( \frac { 7 } { 10 } \right) ^ { \rangle }$  
A) $- \frac { 7 } { 3 } j$ 
B) $\frac { 7 } { 3 }$  
C) $- \frac { 7 } { 10 } j$
D) $\frac { 7 } { 10 }$
E) $- \frac { 7 } { 3 }$

Quizplus · Accepted Answer

The series is a geometric series with the first term $a = \left( \frac{7}{10} \right)^1 = \frac{7}{10}$ and common ratio $r = \frac{7}{10}$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, where $|r| < 1$. Substituting $a = \frac{7}{10}$ and $r = \frac{7}{10}$, we get $S = \frac{\frac{7}{10}}{1 - \frac{7}{10}} = \frac{7}{10} \cdot \frac{10}{3} = \frac{7}{3}$.

Find the Sum of the Infinite Series ∑j=1∞(710)⟩\sum _ { j = 1 } ^ { \infty } \left( \frac { 7 } { 10 } \right) ^ { \rangle }∑j=1∞​(107​)⟩

Find the Sum of the Infinite Series $\sum _ { j = 1 } ^ { \infty } \left( \frac { 7 } { 10 } \right) ^ { \rangle }$