To find the sum of an infinite geometric series $S_\infty$, we use the following formula:

$S_\infty = \frac{a_1}{1 - r}$

Where:

• $S_\infty$ is the sum of the infinite series,
• $a_1$ is the first term of the sequence,
• $r$ is the common ratio.

For the sequence $9, 3, 1, \dots$:

• $a_1 = 9$
• $r = \frac{3}{9} = \frac{1}{3}$

Substituting these values into the formula:

$S_\infty = \frac{9}{1 - \frac{1}{3}} = \frac{9}{\frac{2}{3}} = 9 \times \frac{3}{2} = \frac{27}{2} = 13.5$

So, the sum of the infinite geometric series $S_\infty$ is 13.5.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you determine if an infinite geometric series converges?
2. What is the sum to infinity for a sequence with a common ratio $r$ greater than 1?
3. Can an infinite geometric series have a negative sum?
4. How does the common ratio affect the sum of an infinite geometric series?
5. How would the sum to infinity change if the first term was doubled?

Tip: For an infinite geometric series to converge (i.e., have a finite sum), the common ratio $r$ must satisfy $|r| < 1$.