Let's solve the problem step by step.

The given geometric series is:

$2 + z + \frac{1}{2} z^2 + \dots$

Part (a) - For which $z$ does the series converge?

A geometric series converges if the absolute value of the common ratio $r$ is less than 1:

$|r| < 1$

First, identify the common ratio $r$ of the series. The first term $a$ is 2, and the second term is $z$, so the common ratio $r$ is:

$r = \frac{z}{2}$

For the series to converge:

$\left| \frac{z}{2} \right| < 1$

This simplifies to:

$|z| < 2$

Therefore, the series converges for:

$|z| < 2$

Part (b) - Can the sum of the series found in (a) be $-2$?

To find the sum of a convergent geometric series, use the formula:

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

In this series, the first term $a$ is 2, and the common ratio $r$ is $\frac{z}{2}$. Thus, the sum $S$ of the series is:

$S = \frac{2}{1 - \frac{z}{2}}$

Simplify the denominator:

$S = \frac{2}{\frac{2 - z}{2}} = \frac{2 \cdot 2}{2 - z} = \frac{4}{2 - z}$

We need to determine if this sum can be $-2$:

$\frac{4}{2 - z} = -2$

Solve for $z$:

$4 = -2(2 - z)$ $4 = -4 + 2z$ $8 = 2z$ $z = 4$

However, from part (a), we know that the series converges only for $|z| < 2$. Since $z = 4$ does not satisfy this condition, the series cannot have a sum of $-2$.

Summary

(a) The series converges for $|z| < 2$.

(b) The sum of the series cannot be $-2$ since $z = 4$ is outside the interval of convergence $|z| < 2$.

Do you have any questions or need further details on any part? Here are some related questions you might find interesting:

1. What is the sum of the series when $z = 1$?
2. How does the sum change if the first term $a$ is different?
3. What happens if the common ratio $r$ is greater than 1?
4. Can the series be transformed into another form that is easier to sum?
5. How would you represent the series if $|z| \geq 2$?
6. What are some real-world applications of geometric series?
7. How would you check the convergence of a non-geometric series?
8. What is the radius of convergence for a power series in general?

Tip: Always verify the conditions of convergence before attempting to find the sum of an infinite series.