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The image you uploaded appears to contain a mathematical series:

\[
\frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + \dots
\]

This is an infinite geometric series of the form:

\[
S = \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + \dots
\]

To find the sum of this series, we recognize that it has the first term \( a = \frac{1}{x} \) and the common ratio \( r = \frac{1}{x} \), assuming \( |r| < 1 \), i.e., \( |x| > 1 \).

The sum of an infinite geometric series is given by the formula:

\[
S = \frac{a}{1 - r}
\]

Substituting \( a = \frac{1}{x} \) and \( r = \frac{1}{x} \), we get:

\[
S = \frac{\frac{1}{x}}{1 - \frac{1}{x}} = \frac{\frac{1}{x}}{\frac{x-1}{x}} = \frac{1}{x-1}
\]

Thus, the sum of the series is:

\[
S = \frac{1}{x - 1} \quad \text{for} \quad |x| > 1
\]

Would you like more details on geometric series, or any other questions about this result?

Here are 5 related questions:
1. How can you derive the sum formula for an infinite geometric series?
2. What happens if \( |x| \leq 1 \) in this series?
3. Can this series be applied in real-world problems, such as finance or physics?
4. How do we generalize geometric series for complex numbers?
5. What is the importance of convergence in infinite series?

**Tip**: Always check the conditions for convergence before summing an infinite series to ensure that the sum formula is applicable.

What is the sum of the infinite series: 1/x + 1/x^2 + 1/x^3 + ...?

Learn how to sum the infinite geometric series 1/x + 1/x^2 + 1/x^3 + ... for |x| > 1 using the geometric series sum formula. Explore key concepts such as first terms, common ratios, and convergence conditions. Suitable for high school or college-level mathematics students.

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