Let's analyze the series

$\sum_{n=1}^{\infty} \frac{x^n}{1 + x^n}$

to determine its convergence or divergence depending on the value of $x$.

Series Analysis:

The general term of the series is

$\frac{x^n}{1 + x^n}$

For $x > 1$:

• As $n$ increases, $x^n$ grows larger, making $1 + x^n \approx x^n$.
• Therefore, $\frac{x^n}{1 + x^n} \approx \frac{x^n}{x^n} = 1$.

Since each term of the series approaches 1, the series diverges because the sum of infinitely many 1's diverges.

For $x = 1$:

• When $x = 1$, each term becomes $\frac{1^n}{1 + 1^n} = \frac{1}{2}$.
• The series becomes $\sum_{n=1}^{\infty} \frac{1}{2}$, which diverges since it is a constant non-zero value summed infinitely.

For $0 < x < 1$:

• As $n$ increases, $x^n$ approaches 0 because $x < 1$, making $1 + x^n \approx 1$.
• Therefore, $\frac{x^n}{1 + x^n} \approx \frac{x^n}{1} = x^n$.

This is a geometric series with a common ratio less than 1, which converges.

For $x = 0$:

• When $x = 0$, every term is 0, so the series converges trivially.

Conclusion:

The series converges for $x < 1$ and diverges for $x \geq 1$.

Thus, the correct answer is:

B) converge for $x < 1$, diverge for $x \geq 1$.

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Would you like to explore any details further or have any questions?

Here are some related questions you might find interesting:

1. How does the convergence of geometric series relate to this problem?
2. What is the ratio test, and how could it be applied to this series?
3. How do different values of $x$ affect the behavior of $x^n$ as $n$ increases?
4. What are some examples of oscillatory series, and why doesn't this series fall into that category?
5. How can we classify series in terms of absolute and conditional convergence?

Tip: Understanding how different values of $x$ affect the terms in a series is crucial in determining its convergence or divergence.