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The given question asks about the behavior of the infinite series:

\[
S(x) = \frac{x}{1+x} + \frac{x^2}{1+x^2} + \frac{x^3}{1+x^3} + \cdots
\]

We need to analyze the convergence or divergence of this series based on different values of \( x \).

### Step 1: General Term of the Series
The general term of the series can be written as:

\[
T_n(x) = \frac{x^n}{1+x^n}, \text{ where } n \geq 1.
\]

### Step 2: Behavior of the General Term for Different \( x \)

- **For \( x > 1 \):**
  When \( x > 1 \), as \( n \to \infty \), the term \( \frac{x^n}{1 + x^n} \to 1 \), because \( x^n \) grows very large. Therefore, the series tends to behave like the sum of 1's for large \( n \), which leads to divergence.

- **For \( x = 1 \):**
  When \( x = 1 \), the general term becomes \( \frac{1^n}{1 + 1^n} = \frac{1}{2} \), which is a constant. Summing a constant value repeatedly leads to divergence.

- **For \( 0 < x < 1 \):**
  When \( 0 < x < 1 \), as \( n \to \infty \), the term \( \frac{x^n}{1 + x^n} \to 0 \), because \( x^n \to 0 \) for large \( n \). Thus, the terms become smaller and smaller, suggesting that the series may converge.

- **For \( x = 0 \):**
  When \( x = 0 \), all terms of the series are zero, so the series trivially converges to 0.

### Step 3: Conclusion

- The series **converges** for \( x < 1 \).
- The series **diverges** for \( x \geq 1 \).

### Answer:

The correct option is:
\[
\boxed{(b) \text{Convergent for } x < 1 \text{ and divergent for } x \geq 1.}
\]

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Do you want more details on any specific step or further clarification?

Here are some related questions for deeper understanding:

1. How does the ratio test help in determining the convergence of series?
2. What happens to a series when the general term tends to 0 but not fast enough?
3. Can a series converge if each term tends to 1 as \( n \to \infty \)?
4. How does the behavior of a series change when \( x = 1 \)?
5. What other convergence tests could be applied to this series?

**Tip:** Always consider the behavior of the general term of a series as \( n \to \infty \) when analyzing convergence.

The infinite variable series \( \frac{x}{1+x} + \frac{x^2}{1+x^2} + \frac{x^3}{1+x^3} + \cdots \) is _______.

Explore the convergence and divergence of the infinite series \( \frac{x}{1+x} + \frac{x^2}{1+x^2} + \cdots \). Understand how the series behaves for different values of x and when it converges or diverges. This problem is suitable for university-level students.

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