Find the sum of the infinite series. $$\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 10 } \right) ^ { i }$$
A) $\frac { 2 } { 9 }$
B) $\frac { 9 k } { 2 }$
C) $\frac { 9 } { 2 }$
D) $- \frac { 2 k } { 9 }$
E) $\frac { 2 k } { 9 }$

Question

Find the sum of the infinite series.​ $$\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 10 } \right) ^ { i }$$ ​
A) $\frac { 2 } { 9 }$ ​
B) $\frac { 9 k } { 2 }$ ​
C) $\frac { 9 } { 2 }$ ​
D) $- \frac { 2 k } { 9 }$ ​
E) $\frac { 2 k } { 9 }$ ​

Quizplus · Accepted Answer

This is a geometric series with the first term $a = 2 \times \frac{1}{10} = \frac{2}{10}$ and common ratio $r = \frac{1}{10}$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, so substituting $a = \frac{2}{10}$ and $r = \frac{1}{10}$ gives $S = \frac{\frac{2}{10}}{1 - \frac{1}{10}} = \frac{\frac{2}{10}}{\frac{9}{10}} = \frac{2}{9}$.

Find the Sum of the Infinite Series ∑i=1∞2(110)i\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 10 } \right) ^ { i }i=1∑∞​2(101​)i

Find the Sum of the Infinite Series $\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 10 } \right) ^ { i }$