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.