The relevant content from the PDF includes Problem 4, which involves determining the convergence or divergence of several series. Here’s a summary of the problem components:

Problem 4

Determine whether the following series converge or diverge, and state why. If the series converges, find its sum.

(a) $\sum_{n=1}^{\infty} \frac{1}{1 + 3n^5}$

(b) $\sum_{n=1}^{\infty} \frac{2n}{1 + 4n + 3n^2}$

(c) $\sum_{n=1}^{\infty} \frac{1}{n(n + 1)(n + 3)}$

(d) $\sum_{n=1}^{\infty} e^{-n} \sin\left(\frac{2}{n}\right)$ (Hint: $\lim_{n \to \infty} \frac{\sin\left(\frac{2}{n}\right)}{2/n} = 1$)

(e) $\sum_{n=1}^{\infty} \frac{5^n}{2^n + 2n}$

(f) $\sum_{n=2}^{\infty} \frac{3^n}{n^2 + 9}$

(g) $1 + 0 + \frac{3}{9} + \frac{0}{27} + \ldots$

(h) $\sum_{n=1}^{\infty} \left(\frac{1}{3^n} + \frac{2}{9^n}\right)$

Let's analyze each part for convergence or divergence.

(a) $\sum_{n=1}^{\infty} \frac{1}{1 + 3n^5}$

• Comparison Test: Compare with $\frac{1}{3n^5}$. Since $\sum \frac{1}{n^5}$ converges, so does this series.

(b) $\sum_{n=1}^{\infty} \frac{2n}{1 + 4n + 3n^2}$

• Asymptotic Behavior: As $n \to \infty$, $\frac{2n}{3n^2} = \frac{2}{3n}$. Since $\sum \frac{1}{n}$ diverges, this series diverges.

(c) $\sum_{n=1}^{\infty} \frac{1}{n(n + 1)(n + 3)}$

• Partial Fraction Decomposition: $\frac{1}{n(n + 1)(n + 3)} = \frac{A}{n} + \frac{B}{n + 1} + \frac{C}{n + 3}$. This leads to a telescoping series, which converges.

(d) $\sum_{n=1}^{\infty} e^{-n} \sin\left(\frac{2}{n}\right)$

• Limit Comparison Test: $\lim_{n \to \infty} e^{-n} \sin\left(\frac{2}{n}\right) \approx e^{-n} \cdot \frac{2}{n}$. Since $\sum e^{-n}/n$ converges, this series converges.

(e) $\sum_{n=1}^{\infty} \frac{5^n}{2^n + 2n}$

• Ratio Test: The dominant term $5^n/2^n$ leads to divergence because the series behaves like $\sum (2.5)^n$.

(f) $\sum_{n=2}^{\infty} \frac{3^n}{n^2 + 9}$

• Comparison Test: Since $n^2$ dominates, compare with $\sum \frac{3^n}{n^2}$, which diverges.

(g) $1 + 0 + \frac{3}{9} + \frac{0}{27} + \ldots$

• Geometric Series: The series has terms that converge to zero and effectively diverge due to the nature of the non-zero terms.

(h) $\sum_{n=1}^{\infty} \left(\frac{1}{3^n} + \frac{2}{9^n}\right)$

• Geometric Series: This is a sum of two converging geometric series, both converging to $\frac{1/3}{1-1/3} + \frac{2/9}{1-1/9}$.

Summary of Results

• (a) Converges
• (b) Diverges
• (c) Converges
• (d) Converges
• (e) Diverges
• (f) Diverges
• (g) Diverges
• (h) Converges

Would you like detailed solutions or explanations for any specific part? Here are some related questions:

1. How do you apply the Comparison Test in detail?
2. What is partial fraction decomposition, and how is it used in series?
3. Can you explain the Ratio Test with examples?
4. What conditions determine the convergence of geometric series?
5. How do the behavior of terms in a series affect convergence?

Tip: Always check the behavior of the terms as $n$ approaches infinity to decide on convergence.