22. ∑(k = 0 to ∞)(3/5)^k

Explanation:

Step1: Identify the geometric - series form

This is a geometric series of the form $\sum_{k = 0}^{\infty}ar^{k}$, where $a = 1$ and $r=\frac{3}{5}$.

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

The sum of an infinite geometric series with $|r|\lt1$ is given by $S=\frac{a}{1 - r}$. Here, substituting $a = 1$ and $r=\frac{3}{5}$ into the formula, we get $S=\frac{1}{1-\frac{3}{5}}$.

Step3: Simplify the expression

First, calculate the denominator: $1-\frac{3}{5}=\frac{5 - 3}{5}=\frac{2}{5}$. Then, $S=\frac{1}{\frac{2}{5}}=\frac{5}{2}$.

Answer:

$\frac{5}{2}$