find the sum of the following infinite series: ∑(n = 0 to ∞) 1/2^n\no 2\no 3\no 1/2\no 4

Explanation:

Step1: Identify the series type

The series $\sum_{n = 0}^{\infty}\frac{1}{2^{n}}$ is a geometric - series with the general form $\sum_{n = 0}^{\infty}ar^{n}$, where $a = 1$ (when $n = 0$, $\frac{1}{2^{0}}=1$) and $r=\frac{1}{2}$.

Step2: Apply the formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series is $S=\frac{a}{1 - r}$ when $|r|\lt1$. Here, $a = 1$ and $r=\frac{1}{2}$, so $S=\frac{1}{1-\frac{1}{2}}$.

Step3: Simplify the expression

$S=\frac{1}{\frac{1}{2}}=2$.

Answer:

A. 2