To determine whether the given infinite series converges and to calculate the sum (if it exists), we identify whether the series is a geometric series. The general form of a geometric series is:

$S = a + ar + ar^2 + ar^3 + \cdots$

where $a$ is the first term and $r$ is the common ratio between consecutive terms.

Step 1: Identify $a$ and $r$

From the series: $-3 + 0.6 + (-0.12) + \cdots$

• The first term $a = -3$.
• The common ratio $r$ is calculated as: $r = \frac{\text{second term}}{\text{first term}} = \frac{0.6}{-3} = -0.2.$

Step 2: Check convergence

For a geometric series to converge, the absolute value of the common ratio must satisfy: $|r| < 1.$

Here, $|r| = |-0.2| = 0.2 < 1$, so the series converges.

Step 3: Calculate the sum

The sum of an infinite geometric series is given by: $S = \frac{a}{1 - r}, \quad \text{where } |r| < 1.$

Substituting $a = -3$ and $r = -0.2$: $S = \frac{-3}{1 - (-0.2)} = \frac{-3}{1 + 0.2} = \frac{-3}{1.2}.$

Simplify: $S = -2.5.$

Final Answer:

The sum of the infinite series is: $\boxed{-2.5}$

Would you like a detailed derivation or more related examples? Let me know! Here are 5 follow-up questions and one tip for deeper exploration:

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Related Questions:

1. How does the formula for the sum of an infinite geometric series differ from that of a finite geometric series?
2. What happens if $|r| \geq 1$ in a geometric series?
3. Can a geometric series have both positive and negative terms and still converge?
4. How would you calculate the partial sum of the first $n$ terms of a geometric series?
5. What are some real-world applications of geometric series?

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Tip:

When dealing with infinite series, always test for convergence first. If a series does not converge, its sum cannot be computed!