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x/1+x + x^2/1+x^2 + x^3/1+x^3... 
Check converge x=-2

To check the convergence of the series 

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

at \(x = -2\), we need to evaluate the behavior of each term of the series at \(x = -2\).

### Evaluating the Series at \(x = -2\)

Let's evaluate each term when \(x = -2\):

1. **First term**: \( \frac{x}{1+x} \)  
   \[
   \frac{-2}{1 + (-2)} = \frac{-2}{-1} = 2.
   \]

2. **Second term**: \( \frac{x^2}{1+x^2} \)  
   \[
   \frac{(-2)^2}{1 + (-2)^2} = \frac{4}{1 + 4} = \frac{4}{5}.
   \]

3. **Third term**: \( \frac{x^3}{1+x^3} \)  
   \[
   \frac{(-2)^3}{1 + (-2)^3} = \frac{-8}{1 - 8} = \frac{-8}{-7} = \frac{8}{7}.
   \]

4. **General term**: \( \frac{x^n}{1+x^n} \) at \(x = -2\)  
   \[
   \frac{(-2)^n}{1 + (-2)^n}.
   \]

### Analyzing Convergence

To determine convergence, we analyze the general behavior of the \(n\)-th term as \(n \to \infty\):

- When \(n\) is **even**, \( (-2)^n \) is positive and \( 1 + (-2)^n \) is positive:
  \[
  \frac{(-2)^n}{1 + (-2)^n} = \frac{2^n}{1 + 2^n}.
  \]

  For large \(n\), the fraction \( \frac{2^n}{1 + 2^n} \approx 1 \).

- When \(n\) is **odd**, \( (-2)^n \) is negative, and \( 1 + (-2)^n \) is negative:
  \[
  \frac{(-2)^n}{1 + (-2)^n} = \frac{-2^n}{1 - 2^n}.
  \]

  For large \(n\), the fraction \( \frac{-2^n}{1 - 2^n} \approx 1 \).

### Conclusion on Convergence

In both cases, as \(n \to \infty\), the absolute value of the term \(\frac{(-2)^n}{1 + (-2)^n}\) approaches 1. 

Since the terms of the series do not tend to 0, the series **does not converge**. In fact, the terms oscillate between values near 1 as \(n\) increases, so the series diverges.

### Final Answer

The series \(\frac{x}{1+x} + \frac{x^2}{1+x^2} + \frac{x^3}{1+x^3} + \cdots\) **diverges** when \(x = -2\).

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Would you like more details on this, or do you have any questions?

Here are some related questions to consider:
1. What is the test for divergence of a series?
2. How can you determine if a series converges absolutely or conditionally?
3. What are the different types of convergence for series (e.g., pointwise, uniform)?
4. How does the ratio test apply to this series?
5. How can alternating series converge, and does this series resemble any alternating series?

**Tip:** When analyzing series for convergence, always consider the behavior of the terms as \(n \to \infty\) and check if the terms go to zero.

Learn how to evaluate the convergence of the series x/(1+x) + x^2/(1+x^2) + x^3/(1+x^3)... at x = -2 with detailed step-by-step analysis. This problem covers series convergence, infinite series, and limit of sequences, exploring the behavior of terms as x approaches -2. Suitable for advanced undergraduate students.

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