Question

Explain how to tell if a geometric series is convergent or divergent. Include an example of a convergent series and an example of a divergent series in your explanation.
(4 points)

Answer 1

To determine whether a geometric series converges or diverges, you need to assess the common ratio of the series, denoted as \( r \). A geometric series is of the form:

\[
S = a + ar + ar^2 + ar^3 + \ldots
\]

where \( a \) is the first term and \( r \) is the common ratio.

### Convergence and Divergence Criteria:
- **Convergent:** A geometric series converges if the absolute value of the common ratio is less than 1, \( |r| < 1 \). When a geometric series converges, the sum can be calculated using the formula:

\[
S = \frac{a}{1 - r}
\]

- **Divergent:** A geometric series diverges if the absolute value of the common ratio is greater than or equal to 1, \( |r| \geq 1 \). In this case, the series does not approach a finite limit.

### Examples:

1. **Example of a Convergent Series:**
Consider the series:

\[
S = 2 + 2 \cdot \frac{1}{2} + 2 \cdot \left(\frac{1}{2}\right)^2 + 2 \cdot \left(\frac{1}{2}\right)^3 + \ldots
\]

Here, \( a = 2 \) and the common ratio \( r = \frac{1}{2} \).

Since \( |r| = \frac{1}{2} < 1 \), the series converges. We can find its sum using the formula:

\[
S = \frac{2}{1 - \frac{1}{2}} = \frac{2}{\frac{1}{2}} = 4
\]

Therefore, this geometric series converges to 4.

2. **Example of a Divergent Series:**
Consider the series:

\[
S = 3 + 3 \cdot 2 + 3 \cdot 2^2 + 3 \cdot 2^3 + \ldots
\]

Here, \( a = 3 \) and the common ratio \( r = 2 \).

Since \( |r| = 2 \geq 1 \), the series diverges. This means the terms continue to grow without approaching a finite limit.

### Summary:
- For a geometric series, check the common ratio \( r \):
- If \( |r| < 1 \), the series converges.
- If \( |r| \geq 1 \), the series diverges.