To determine whether the given series is convergent or divergent, we need to recognize it as a geometric series and analyze its common ratio.

Step 1: Identify the series

The given series is:

$6 + 5 + \frac{25}{6} + \frac{125}{36} + \dots$

This is a geometric series, where each term can be written in the form $a \cdot r^{n-1}$, with $a$ being the first term and $r$ the common ratio.

Step 2: Find the common ratio $r$

To find the common ratio $r$, divide the second term by the first term:

$r = \frac{5}{6}$

Similarly, divide the third term by the second term:

$r = \frac{\frac{25}{6}}{5} = \frac{25}{30} = \frac{5}{6}$

This shows that the common ratio $r$ is $\frac{5}{6}$.

Step 3: Determine convergence

A geometric series converges if the absolute value of the common ratio $r$ is less than 1, i.e., $|r| < 1$.

Since $|r| = \frac{5}{6} < 1$, the series is convergent.

Step 4: Find the sum

The sum $S$ of an infinite geometric series is given by:

$S = \frac{a}{1 - r}$

where $a$ is the first term and $r$ is the common ratio. Here, $a = 6$ and $r = \frac{5}{6}$.

$S = \frac{6}{1 - \frac{5}{6}} = \frac{6}{\frac{1}{6}} = 6 \times 6 = 36$

Conclusion:

The series converges, and the sum is $\boxed{36}$.

Would you like more details or have any questions?

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Here are five related questions:

1. How do you determine the common ratio in a geometric series?
2. What is the general condition for a geometric series to converge?
3. How would the sum change if the first term was different?
4. Can a geometric series with a common ratio greater than 1 ever converge?
5. How does the sum of a geometric series relate to its first term and common ratio?

Tip: When dealing with geometric series, always check the absolute value of the common ratio. If it's less than 1, the series will converge, and you can find its sum easily using the formula.