To solve the given infinite geometric series, we follow these steps:

The series is: $6 + \frac{6}{7} + \frac{6}{7^2} + \frac{6}{7^3} + \dots$

Step 1: Recognize the structure of the series

This is a geometric series with:

• First term ($a$) = $6$,
• Common ratio ($r$) = $\frac{1}{7}$, since each term is multiplied by $\frac{1}{7}$.

Step 2: Use the formula for the sum of an infinite geometric series

The sum of an infinite geometric series is given by: $S = \frac{a}{1 - r}$ where $|r| < 1$.

Step 3: Substitute the values

$S = \frac{6}{1 - \frac{1}{7}} = \frac{6}{\frac{6}{7}} = 6 \cdot \frac{7}{6} = 7$

Final Answer:

The sum of the infinite geometric series is 7.

Let me know if you'd like further clarification or steps!

1. What are the conditions for a geometric series to have a finite sum?
2. How does the common ratio affect the convergence of the series?
3. Can a geometric series diverge? Under what conditions?
4. What happens if the first term is negative? Does it change the process?
5. How can we apply this to practical problems, such as calculating compound interest?

Tip: Always check the common ratio to ensure $|r| < 1$, which guarantees convergence of the series.