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This expression represents a repeating decimal series with specific coefficients in each power of \(10\). Let’s solve it step by step:

### Given Series:
\[
1 + \frac{3}{10} + \frac{7}{10^2} + \frac{9}{10^3} + \frac{3}{10^4} + \frac{7}{10^5} + \frac{9}{10^6} + \ldots
\]

1. **Pattern Identification**:
   - The repeating coefficients are: \(3, 7, 9\), and they cycle every 3 terms.
   - This series can be written in a compact way using powers of \(10\):
     \[
     S = 1 + \left(\frac{3}{10} + \frac{7}{10^2} + \frac{9}{10^3}\right) + \left(\frac{3}{10^4} + \frac{7}{10^5} + \frac{9}{10^6}\right) + \ldots
     \]

2. **Group and Simplify**:
   - Factor out the repeating block \( \frac{3}{10} + \frac{7}{10^2} + \frac{9}{10^3} \):
     \[
     S = 1 + \left(\frac{3}{10} + \frac{7}{10^2} + \frac{9}{10^3}\right) \left(1 + \frac{1}{10^3} + \frac{1}{10^6} + \ldots\right)
     \]

3. **Simplify Infinite Geometric Series**:
   - The repeating block sum is:
     \[
     \frac{3}{10} + \frac{7}{10^2} + \frac{9}{10^3} = \frac{3}{10} + \frac{7}{100} + \frac{9}{1000} = \frac{300 + 70 + 9}{1000} = \frac{379}{1000}.
     \]
   - The second term is an infinite geometric series:
     \[
     1 + \frac{1}{10^3} + \frac{1}{10^6} + \ldots = \frac{1}{1 - \frac{1}{10^3}} = \frac{1}{1 - 0.001} = \frac{1}{0.999} = 1.001.
     \]

4. **Combine Everything**:
   - Substitute back:
     \[
     S = 1 + \frac{379}{1000} \times 1.001.
     \]
   - Multiply:
     \[
     \frac{379}{1000} \times 1.001 = \frac{379.379}{1000}.
     \]
   - Add to \(1\):
     \[
     S = 1 + 0.379379 = 1.379379.
     \]

### Final Answer:
\[
S = 1.379379\ldots = 1.\overline{379}.
\]

Would you like more details or a verification of this result?

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### Related Questions:
1. How do geometric series help in solving infinite decimal expansions?
2. Can you explain why the repeating pattern contributes to a rational number?
3. What other forms can \(1.\overline{379}\) be expressed as?
4. Can this result be verified by converting \(1.\overline{379}\) back into a fraction?
5. How would the result change if the repeating block were longer?

### Tip:
Always look for repeating patterns and group terms systematically to simplify infinite series problems!

Evaluate the infinite series: 1 + 3/10 + 7/10^2 + 9/10^3 + 3/10^4 + 7/10^5 + 9/10^6 + ...

Explore how to evaluate the infinite series 1 + 3/10 + 7/10^2 + 9/10^3 + ... by identifying repeating decimal patterns and applying geometric series formulas. This problem is ideal for high school students interested in advanced algebra concepts.

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