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

Question

Quizplus · Accepted Answer

The series is a geometric series with first term $ a = 2 \times \frac{1}{4} = \frac{1}{2} $ and common ratio $ r = \frac{1}{4} $. The sum of an infinite geometric series is given by $ \frac{a}{1-r} $. Substituting the values, we get $ \frac{\frac{1}{2}}{1-\frac{1}{4}} = \frac{\frac{1}{2}}{\frac{3}{4}} = \frac{2}{3} $.

Find the Sum of the Infinite Series ∑i=1∞2(14)i\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 4 } \right) ^ { i }i=1∑∞​2(41​)i

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