Find the sum of the infinite series. $$\sum _ { j = 1 } ^ { \infty } \left( \frac { 7 } { 10 } \right) ^ { j }$$
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.​ $$\sum _ { j = 1 } ^ { \infty } \left( \frac { 7 } { 10 } \right) ^ { j }$$ ​
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

This 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 the given values, 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)j\sum _ { j = 1 } ^ { \infty } \left( \frac { 7 } { 10 } \right) ^ { j }j=1∑∞​(107​)j

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