Given the sequence 9,3,1…, find S infinity

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 \( |r| < 1 \).

Learn how to find the sum of an infinite geometric series for the sequence 9, 3, 1... with a step-by-step solution. This problem covers key concepts like infinite geometric series, common ratio, and the formula for the sum of an infinite series. Suitable for high school students.

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