find the sum of the following infinite series: ∑(n = 0 to ∞) 1/4^n

Explanation:

Step1: Identify the series type

The series $\sum_{n = 0}^{\infty}\frac{1}{4^{n}}$ is a geometric - series with the general form $\sum_{n=0}^{\infty}ar^{n}$, where $a = 1$ (when $n = 0$, $\frac{1}{4^{0}}=1$) and $r=\frac{1}{4}$.

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, since $|r|=\left|\frac{1}{4}\right|=\frac{1}{4}\lt1$, we substitute $a = 1$ and $r=\frac{1}{4}$ into the formula. So $S=\frac{1}{1-\frac{1}{4}}$.

Step3: Simplify the expression

$S=\frac{1}{\frac{4 - 1}{4}}=\frac{1}{\frac{3}{4}}=\frac{4}{3}$.

Answer:

$\frac{4}{3}$