Find the sum of the series.
-$$\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 2 ^ { n } } - \frac { 1 } { 5 ^ { n } } \right)$$
A) $\frac { 7 } { 4 }$
B) $- \frac { 7 } { 4 }$
C) $- \frac { 3 } { 4 }$
D) $\frac { 3 } { 4 }$

Question

Quizplus · Accepted Answer

The series can be split into two separate geometric series: $\sum_{n=1}^{\infty} \frac{1}{2^n}$ and $\sum_{n=1}^{\infty} \frac{1}{5^n}$. The sum of the first series is $\frac{1/2}{1-1/2} = 1$ and the sum of the second series is $\frac{1/5}{1-1/5} = \frac{1}{4}$. Subtracting these gives $1 - \frac{1}{4} = \frac{3}{4}$.

Find the Sum of the Series ∑n=1∞(12n−15n)\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 2 ^ { n } } - \frac { 1 } { 5 ^ { n } } \right)n=1∑∞​(2n1​−5n1​)

Find the Sum of the Series $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 2 ^ { n } } - \frac { 1 } { 5 ^ { n } } \right)$