Find the sum of the series.
-$$\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 2 ^ { n } } - \frac { 1 } { 4 ^ { n } } \right)$$
A) $- 2$
B) $- \frac { 2 } { 3 }$
C) 2
D) $\frac { 2 } { 3 }$

Question

Quizplus · Accepted Answer

The given series can be split into two separate geometric series: $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { 2 ^ { n } }$ and $-\sum _ { n = 0 } ^ { \infty } \frac { 1 } { 4 ^ { n } }$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. For the first series, $a = 1$ and $r = \frac{1}{2}$, giving a sum of $2$. For the second series, $a = 1$ and $r = \frac{1}{4}$, giving a sum of $\frac{4}{3}$. Subtracting the second sum from the first gives $2 - \frac{4}{3} = \frac{2}{3}$.

Find the Sum of the Series ∑n=0∞(12n−14n)\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 2 ^ { n } } - \frac { 1 } { 4 ^ { n } } \right)n=0∑∞​(2n1​−4n1​)

Find the Sum of the Series $\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 2 ^ { n } } - \frac { 1 } { 4 ^ { n } } \right)$