https://file.aiquickdraw.com/mathbot/file/1728939194565uc0hby6p.jpg

This image contains a geometric series:

\[
6 + 0.6 + 0.06 + 0.006 + \dots
\]

This is a geometric series where the first term \( a = 6 \) and the common ratio \( r = 0.1 \) (since each term is 1/10 of the previous one).

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

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

where \( |r| < 1 \).

Substituting the given values:

\[
S = \frac{6}{1 - 0.1} = \frac{6}{0.9} = \frac{60}{9} = 6.6667
\]

Thus, the sum of the convergent series is approximately:

\[
S = 6.6667
\]

Would you like more details on the steps, or do you have any specific questions?

Here are 5 related questions:
1. How does the value of \( r \) affect the convergence of a geometric series?
2. What happens if the common ratio \( r \geq 1 \)?
3. Can you derive the formula for the sum of an infinite geometric series?
4. What if the first term \( a \) were different? How would it change the sum?
5. How do we know this series is geometric and not arithmetic?

**Tip:** Always check if the common ratio in a geometric series satisfies \( |r| < 1 \), as this guarantees the series converges.

Find the sum of the convergent series: 6 + 0.6 + 0.06 + 0.006 + ...

Learn how to find the sum of the convergent geometric series 6 + 0.6 + 0.06 + 0.006 + ... using the geometric series formula. This problem explores the concept of convergent series and is suitable for students in Grades 10-12.

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