Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 } { 7 } \right) ^ { n }$
A)Diverges.
B)Converges to $\frac { 5 } { 2 }$
C)Converges to $\frac { 5 } { 12 }$
D)Converges to $\frac { 7 } { 2 }$

Question

Quizplus · Accepted Answer

The series is a geometric series with a common ratio $r = \frac{5}{7}$, which is less than 1 in absolute value, so the series converges. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, where $a$ is the first term. Here, $a = \left(\frac{5}{7}\right)^1 = \frac{5}{7}$, so the sum is $S = \frac{\frac{5}{7}}{1 - \frac{5}{7}} = \frac{\frac{5}{7}}{\frac{2}{7}} = \frac{5}{2}$.

Determine Whether the Given Geometric Series Converges,and If So,find Its ∑n=1∞(57)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 } { 7 } \right) ^ { n }∑n=1∞​(75​)n

Determine Whether the Given Geometric Series Converges,and If So,find Its $\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 } { 7 } \right) ^ { n }$