I. The following three problems are related.

1. (a) Determine if the series \(\displaystyle \sum_{n=1}^{\infty} (-1)^{n} \frac{1}{\sqrt[3]{n}}\) is convergent or divergent.
2. (b) Determine if the series \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n}}\) is convergent or divergent.
3. (c) From parts (a) and (b) conclude if the series \(\displaystyle \sum_{n=1}^{\infty} (-1)^{n} \frac{1}{\sqrt[3]{n}}\) is absolutely convergent, conditionally convergent, or divergent.

II. Determine whether each of the following series is absolutely convergent, conditionally convergent, or divergent.

1. \(\displaystyle \sum_{n=1}^{\infty} \frac{\sin n}{n^2}\)
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2. \(\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{2n+3}\)
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3. (a) \(\displaystyle \sum_{n=1}^{\infty} (-1)^{n} \frac{1}{n^3}\)
   (b) \(\displaystyle \sum_{n=1}^{\infty} (-1)^{n} \frac{1}{\sqrt[3]{n}}\)
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4. \(\displaystyle \sum_{n=1}^{\infty} (-1)^{n+1} \frac{2^n-1}{3^n-1}\)
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5. \(\displaystyle \sum_{n=1}^{\infty} \frac{(-2)^n}{(2n+1)!}\)
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6. \(\displaystyle \sum_{n=1}^{\infty} \frac{n!}{100^n}\)
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7. \(\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1} \frac{2n+5}{3n+1}\)
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III. Determine whether each of the following series is convergent or divergent.

1. \(\displaystyle \sum_{n=1}^{\infty} \frac{(-2)^n}{n}\)
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2. \(\displaystyle \sum_{n=1}^{\infty} \frac{(n+1) \, 3^n}{2^n \, n^3}\)
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3. \(\displaystyle \sum_{k=1}^{\infty} \frac{3}{k!}\)
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4. \(\displaystyle \sum_{n=1}^{\infty} \left( \frac{n^2+1}{3n^2+2} \right)^n\)
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5. \(\displaystyle \sum_{m=1}^{\infty} \frac{m}{e^m}\)
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6. \(\displaystyle \sum_{n=1}^{\infty} \left( \frac{5n^2-2}{4n^2+3} \right)^n\)
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7. \(\displaystyle \sum_{n=1}^{\infty} \frac{n!}{n^n}\)
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8. \(\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1} \frac{7^{n}}{5^n \, n^3}\)
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9. \(\displaystyle \sum_{n=3}^{\infty} 4 \left(1 + \frac{1}{n}\right)^{n^2}\)
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