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.