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To determine if the series
\[
\sum_{k=0}^{\infty} \frac{(-1)^k}{1 + e^k}
\]
is convergent or divergent, we need to apply tests for series convergence, particularly tests suitable for alternating series, as this series alternates due to the term \( (-1)^k \).

### Step 1: Alternating Series Test (Leibniz Test)
The Alternating Series Test states that an alternating series of the form \(\sum_{k=0}^{\infty} (-1)^k a_k\) converges if the sequence \( a_k \) satisfies two conditions:
1. The terms \( a_k \) are decreasing, i.e., \( a_{k+1} \leq a_k \) for all \( k \).
2. The limit of the terms approaches zero, i.e., \( \lim_{k \to \infty} a_k = 0 \).

Here, the sequence \( a_k = \frac{1}{1 + e^k} \).

### Step 2: Verify the Conditions
1. **Decreasing Terms**:
   The function \( f(k) = \frac{1}{1 + e^k} \) is decreasing because \( e^k \) increases as \( k \) increases, making the denominator larger and the overall fraction smaller.

2. **Limit Approaches Zero**:
   \[
   \lim_{k \to \infty} \frac{1}{1 + e^k} = \frac{1}{1 + \infty} = 0.
   \]
   Therefore, \( \lim_{k \to \infty} a_k = 0 \).

Since both conditions of the Alternating Series Test are satisfied, the series converges.

### Conclusion
The series \(\sum_{k=0}^{\infty} \frac{(-1)^k}{1 + e^k}\) is convergent by the Alternating Series Test.

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

Here are five related questions to expand your understanding:
1. What is the difference between absolute and conditional convergence?
2. How does the ratio test work for determining convergence of a series?
3. Can an alternating series converge even if it fails the Alternating Series Test?
4. What happens if we remove the alternating factor \( (-1)^k \) from the given series?
5. How does the comparison test for series work?

**Tip:** When testing an alternating series, always start with the Alternating Series Test. If it passes, then the series converges, at least conditionally.

Determine if the series ∑_{k=0}^{∞} (-1)^k / (1 + e^k) is convergent or divergent.

Explore how to determine whether the alternating series ∑_{k=0}^{∞} (-1)^k / (1 + e^k) converges or diverges using the Alternating Series Test. This solution is suitable for undergraduate mathematics students and covers key series convergence concepts.

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