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We are tasked with expressing the sum \( \frac{5}{3} + \frac{5}{6} + \frac{5}{12} + \frac{5}{24} + \dots \) as a series of the form \( \sum_{n=0}^{\infty} a_n \) or \( \sum_{n=1}^{\infty} a_n \), and then determining if the series converges.

### Step 1: Identify the pattern

We observe that the denominators of the terms form a sequence: \( 3, 6, 12, 24, \dots \). This looks like a geometric progression where each denominator is doubled in the subsequent term.

- The first term is \( \frac{5}{3} \).
- The second term is \( \frac{5}{6} = \frac{5}{3 \cdot 2} \).
- The third term is \( \frac{5}{12} = \frac{5}{3 \cdot 2^2} \).
- The fourth term is \( \frac{5}{24} = \frac{5}{3 \cdot 2^3} \).

Hence, the general term for this series can be written as:
\[
a_n = \frac{5}{3 \cdot 2^n}, \quad n \geq 0
\]

Thus, the series can be expressed as:
\[
\sum_{n=0}^{\infty} \frac{5}{3 \cdot 2^n}
\]

### Step 2: Simplify and find the sum of the series

This is a geometric series of the form:
\[
\sum_{n=0}^{\infty} \frac{5}{3} \cdot \left( \frac{1}{2} \right)^n
\]

The sum of an infinite geometric series \( \sum_{n=0}^{\infty} ar^n \) is given by:
\[
S = \frac{a}{1 - r} \quad \text{if} \quad |r| < 1
\]

Here, \( a = \frac{5}{3} \) and \( r = \frac{1}{2} \). Therefore, the sum of the series is:
\[
S = \frac{\frac{5}{3}}{1 - \frac{1}{2}} = \frac{\frac{5}{3}}{\frac{1}{2}} = \frac{5}{3} \cdot 2 = \frac{10}{3}
\]

### Conclusion:

The series converges, and the sum is:
\[
S = \frac{10}{3}
\]

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. What conditions are necessary for a geometric series to converge?
2. How do you determine the common ratio in a geometric series?
3. What is the formula for the sum of a geometric series with a given first term and ratio?
4. If the first term of the series was \( \frac{7}{4} \) instead of \( \frac{5}{3} \), how would the sum change?
5. Can you derive the sum of a geometric series where the ratio is negative?

**Tip**: When analyzing infinite series, always check whether the common ratio lies within the interval \( (-1, 1) \) for the series to converge.

Express the sum 5/3 + 5/6 + 5/12 + 5/24 + ... as a series of the form sum(an) and find the sum of the series if it converges.

Learn how to express and solve the geometric series 5/3 + 5/6 + 5/12 + 5/24 + ... by finding its general term and determining the sum of the series if it converges. This problem introduces geometric progressions and the sum of infinite series, suitable for Grades 10-12.

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