Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 1 } ^ { \infty } \left( - \frac { 5 } { 9 } \right) ^ { n }$
A)Converges to $- \frac { 5 } { 14 }$
B)Converges to $\frac { 5 } { 4 }$
C)Diverges
D)Converges to $\frac { 9 } { 14 }$

Question

Quizplus · Accepted Answer

The series is a geometric series with a common ratio $r = -\frac{5}{9}$, which is between -1 and 1 ($-1 < r < 1$), 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, $a = \left(-\frac{5}{9}\right)^1 = -\frac{5}{9}$, so the sum is $S = \frac{-\frac{5}{9}}{1 - (-\frac{5}{9})} = \frac{-\frac{5}{9}}{\frac{14}{9}} = -\frac{5}{14}$.

Determine Whether the Given Geometric Series Converges,and If So,find Its ∑n=1∞(−59)n\sum _ { n = 1 } ^ { \infty } \left( - \frac { 5 } { 9 } \right) ^ { n }∑n=1∞​(−95​)n

Determine Whether the Given Geometric Series Converges,and If So,find Its $\sum _ { n = 1 } ^ { \infty } \left( - \frac { 5 } { 9 } \right) ^ { n }$