Let's analyze the infinite geometric series given in the image:

$1 + \left(\frac{2x}{3}\right) + \left(\frac{2x}{3}\right)^2 + \left(\frac{2x}{3}\right)^3 + \dots$

(a) For what values of $x$ does the series converge?

This is a geometric series of the form:

$S = 1 + r + r^2 + r^3 + \dots$

where $r = \frac{2x}{3}$. A geometric series converges if the absolute value of the common ratio $r$ is less than 1:

$|r| < 1$

Substituting $r = \frac{2x}{3}$:

$\left| \frac{2x}{3} \right| < 1$

To find the range of $x$, we solve:

$\frac{|2x|}{3} < 1$

Multiplying both sides by 3:

$|2x| < 3$

Dividing by 2:

$|x| < \frac{3}{2}$

Thus, the series converges when:

$- \frac{3}{2} < x < \frac{3}{2}$

(b) Find the sum of the series if $x = 1.2$.

The sum $S$ of an infinite geometric series is given by the formula:

$S = \frac{a}{1 - r}$

where $a$ is the first term (which is 1 in this case) and $r$ is the common ratio. Here, $r = \frac{2x}{3}$, so for $x = 1.2$:

$r = \frac{2(1.2)}{3} = 0.8$

Now, applying the formula:

$S = \frac{1}{1 - 0.8} = \frac{1}{0.2} = 5$

Thus, the sum of the series when $x = 1.2$ is $S = 5$.

Would you like more details on any of these steps? Here are 5 related questions to explore this concept further:

1. How does the common ratio affect the convergence of a geometric series?
2. What happens to the sum if $x$ approaches $\frac{3}{2}$?
3. Can you explain the derivation of the sum formula for a geometric series?
4. How would the series change if the first term wasn't 1?
5. What would be the sum if $x = -1.4$?

Tip: When dealing with geometric series, always verify the condition for convergence before attempting to find the sum.