https://r2.erweima.ai/mathbot/image/1726528907791qywf6xsj.jpg

The expression in the image represents an infinite geometric series:

\[
\left( \frac{1}{3} \right) + \left( \frac{1}{3} \right)^2 + \left( \frac{1}{3} \right)^3 + \cdots
\]

This is an infinite geometric series where the first term \( a = \frac{1}{3} \) and the common ratio \( r = \frac{1}{3} \).

The sum \( S \) of an infinite geometric series can be calculated using the formula:

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

Since \( a = \frac{1}{3} \) and \( r = \frac{1}{3} \), we can substitute these values into the formula:

\[
S = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2}
\]

Thus, the sum of the series is:

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

Would you like more details on geometric series, or do you have any other questions?

Here are 5 related questions you might find interesting:
1. How do we determine if a geometric series converges or diverges?
2. What happens if the common ratio \( r \) is greater than 1?
3. Can you find the sum of a finite geometric series?
4. How does the formula for the sum of an infinite geometric series change if \( r \) is negative?
5. Can a geometric series have complex numbers as terms?

**Tip:** A geometric series converges only if the absolute value of the common ratio \( r \) is less than 1 (i.e., \( |r| < 1 \)).

What is the sum of the infinite geometric series (1/3) + (1/3)^2 + (1/3)^3 + ...?

Learn how to calculate the sum of the infinite geometric series (1/3) + (1/3)^2 + (1/3)^3 + ... using the formula for the sum of an infinite geometric series. In this example, the first term is 1/3, and the common ratio is 1/3. This problem is suitable for high school students in Grades 9-12.

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