The sum of an infinite geometric series can be found using the formula:

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

where:

• $a$ is the first term of the series,
• $r$ is the common ratio of the series (the factor by which each term is multiplied to get the next term), and
• $|r| < 1$ for the series to converge.

Step 1: Identify the first term $a$

The series given is:

$9 - 3 - 1 - \frac{1}{3} + \cdots$

Here, the first term $a$ is 9.

Step 2: Find the common ratio $r$

We can find the common ratio by dividing the second term by the first term:

$r = \frac{-3}{9} = -\frac{1}{3}$

Thus, the common ratio $r$ is $-\frac{1}{3}$.

Step 3: Use the sum formula

Now that we know $a = 9$ and $r = -\frac{1}{3}$, we can substitute these values into the formula:

$S = \frac{9}{1 - \left(-\frac{1}{3}\right)} = \frac{9}{1 + \frac{1}{3}} = \frac{9}{\frac{4}{3}} = 9 \times \frac{3}{4} = \frac{27}{4}$

Final Answer:

The sum of the infinite geometric series is:

$S = \frac{27}{4}$

Would you like further details or clarification on this?

Here are five related questions:

1. How do you find the common ratio in a geometric series?
2. What condition must be met for the sum of an infinite geometric series to exist?
3. What happens to the sum formula if $r \geq 1$?
4. How can you find the sum of a finite geometric series?
5. How does the common ratio affect the convergence of the series?

Tip: Always ensure that the common ratio $|r| < 1$ before applying the sum formula for an infinite series!