The given series is:

$\frac{x}{1+x} - \frac{x^2}{1+x^2} + \frac{x^3}{1+x^3} - \cdots$

where $0 < x < 1$.

To determine if this series is convergent or divergent, let's analyze the behavior of the terms.

1. Observing Terms: Each term of the series is of the form $\frac{x^n}{1 + x^n}$, and the terms alternate in sign. For $0 < x < 1$, as $n$ becomes large, $x^n$ approaches 0 because $x$ is a fraction, which causes each term $\frac{x^n}{1 + x^n}$ to approach 0.

2. Convergence Criteria: Since this series has alternating terms and the absolute value of each term $\frac{x^n}{1 + x^n}$ tends toward 0, we can apply the Alternating Series Test (also known as the Leibniz criterion) for convergence:

   • The terms decrease in magnitude as $n$ increases.
   • The limit of each term as $n \to \infty$ is 0.

   Both conditions are satisfied.

3. Conclusion: By the Alternating Series Test, the series is convergent.

Would you like a more detailed breakdown of each step?

Here are some related questions for deeper exploration:

1. How does the Alternating Series Test work, and why does it imply convergence?
2. What happens if $x$ is close to 1? Does it impact the convergence rate?
3. Can we find the sum of this series for specific values of $x$?
4. Is there an alternative test that could confirm convergence for this series?
5. How would the convergence behavior change if the terms did not alternate in sign?

Tip: In alternating series, if the terms decrease in magnitude to zero, it’s generally convergent by the Alternating Series Test.