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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.

Explore how to determine the convergence and summation of a geometric series problem step-by-step. Learn about geometric series, convergence criteria, and the formula for summing series. Suitable for advanced high school students studying calculus and series summation.

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