This is a Calculus II Problem Set focused on series tests and convergence, covering topics such as the Divergence Test, p-test, Comparison Tests, and applications of series. Here's a breakdown of the problem set you provided:

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1. Divergence Test:

You are asked to determine if the following series diverge using the Divergence Test or if the test is inconclusive:

• $\sum_{k=0}^{\infty} \sqrt{k}$
• $\sum_{k=0}^{\infty} \frac{k^3}{k!}$
• $\sum_{k=2}^{\infty} \frac{1}{k \ln(k)}$
• $\sum_{k=1}^{\infty} \left( \left( 1 + \frac{1}{k} \right)^k - 1 \right)$

2. Convergence/Divergence:

You need to determine whether the following series converge or diverge using appropriate tests:

• $\sum_{k=2}^{\infty} \frac{1}{k^{\sqrt{2}-1}}$
• $\sum_{k=1}^{\infty} \frac{\sqrt{k}}{k^3 + 1}$
• $\sum_{k=0}^{\infty} \frac{2^{k+2} + k^2}{3^k + k + 12}$
• $\sum_{k=3}^{\infty} \frac{1}{k^2 \ln(k)}$

3. Sine Series:

You are asked to use the Limit Comparison Test to determine if these sine-related series converge or diverge:

• $\sum_{k=1}^{\infty} \sin\left(\frac{1}{k}\right)$
• $\sum_{k=1}^{\infty} \frac{1}{k} \sin\left(\frac{1}{k}\right)$

4. Limit Comparison Test:

You must prove that if $\sum a_k$ converges, then $\sum a_k^2$ also converges, using the Limit Comparison Test.

5. Gabriel’s Wedding Cake:

You are asked to determine whether the volume of a wedding cake of infinite height is finite or infinite by justifying your answer with a series.

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Would you like help with a specific problem or a general explanation of the methods to approach these problems? Here are some questions to further guide your understanding:

1. Do you need help understanding the Divergence Test for a specific series in problem 1?
2. Would you like a detailed explanation of how to apply the Limit Comparison Test in problem 3 or 4?
3. Are you unsure which convergence test to use for one of the series in problem 2?
4. Do you need clarification on how to calculate the volume of Gabriel’s Wedding Cake using series in problem 5?
5. Would you like to see step-by-step solutions for any specific series?

Tip: When using the Divergence Test, if the limit of the general term of the series as $k \to \infty$ does not tend to zero, the series diverges. If it does tend to zero, the test is inconclusive, and other tests must be used.