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The given series is:

\[
\sum_{k=0}^{\infty} \frac{(-1)^k \cdot 4}{k^4 + 5}
\]

This is an alternating series due to the \((-1)^k\) term. To determine if the series converges, we can apply the **Alternating Series Test (Leibniz's Test)**. The test has two conditions:

1. The terms \(a_k = \frac{4}{k^4 + 5}\) must be positive and **decreasing** in magnitude.
2. \(\lim_{k \to \infty} a_k = 0\).

### Step 1: Check if \(a_k = \frac{4}{k^4 + 5}\) is decreasing
As \(k\) increases, \(k^4 + 5\) increases, so \(a_k\) becomes smaller. Hence, \(a_k\) is decreasing for large \(k\).

### Step 2: Check if \(\lim_{k \to \infty} a_k = 0\)
\[
\lim_{k \to \infty} \frac{4}{k^4 + 5} = 0
\]
Since the denominator grows without bound, the limit of \(a_k\) as \(k \to \infty\) is indeed 0.

### Conclusion:
The series meets the conditions of the Alternating Series Test and therefore **converges**.

### Correct Choice:
The correct answer is **D**. The series converges because \(a_k\) is non-increasing in magnitude for \(k\) greater than some index \(N\) and:

\[
\lim_{k \to \infty} a_k = 0
\]

So, you should fill in:

- \(a_k = \frac{4}{k^4 + 5}\)
- \(\lim_{k \to \infty} a_k = 0\)

Do you have any questions or would you like more details?

### Relative questions:
1. What is the Alternating Series Test?
2. How do you prove that a sequence is decreasing?
3. Can the series converge if \(a_k\) doesn't approach zero?
4. How does the behavior of the denominator affect the convergence of the series?
5. What other tests can be used to determine the convergence of a series?

### Tip:
For alternating series, always check if the terms decrease and approach zero. This ensures convergence by the Leibniz's test.

Determine whether the following series converges: ∑ (from k=0 to ∞) [((-1)^k * 4) / (k^4 + 5)]. Select the correct choice below and fill in the answer box(es).

This problem explores the convergence of an alternating series with the form ∑ ((-1)^k * 4) / (k^4 + 5). Using the Alternating Series Test, we determine if the series converges by checking if the terms decrease and approach zero. Ideal for students in Grades 11-12 studying advanced calculus.

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