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Let's go through the problems one by one and analyze them.

### 1. Convergence or Divergence: 
We need to determine whether the following series converge or diverge.

**(a)** \[
\sum_{n=1}^{\infty} \left( 1 + \frac{1}{n} \right)^n
\]
This series involves a term that resembles a binomial expansion and will grow as \(n\) increases. We will analyze the behavior of the term and compare it to a known convergent or divergent series.

**(b)** \[
\sum_{n=1}^{\infty} \left( 1 - \frac{1}{n} \right)^n
\]
Similar to (a), this series also has an expression involving powers of a term. We will evaluate its limit as \(n\) approaches infinity and assess its convergence or divergence.

### 2. Series with Positive Terms:
Let \(\sum_{n=1}^{\infty} a_n\) be a series with positive terms and define:
- \( r_n = \frac{a_{n+1}}{a_n} \)
- \( R_n = a_{n+1} + a_{n+2} + a_{n+3} + \dots \), and \(\lim_{n\to\infty} r_n = L < 1 \)

**(a)** Prove that if \(r_n\) is decreasing, then \( R_n \leq \frac{a_{n+1}}{1 - r_{n+1}} \).

**(b)** Prove that if \(r_n\) is increasing, then \( R_n \leq \frac{a_{n+1}}{1 - L} \).

This involves using properties of decreasing and increasing sequences and comparison with geometric series.

### 3. Use the results from Problem 2 to find an upper bound for the errors when using \(S_5\) as an estimate for \(S_\infty\):
We need to estimate the sums:
**(a)** \(\sum_{n=1}^{\infty} \frac{1}{n 2^n}\)
**(b)** \(\sum_{n=1}^{\infty} \frac{n}{2^n}\)

### 4. Recursively defined sequence:
The sequence \(a_n\) is defined as \(a_1 = 1\) and \(a_n = \frac{2n + 1}{3n - 1} a_{n-1}\) for \(n > 1\).

**(a)** Find \(a_2\), \(a_3\), and \(a_4\).

**(b)** Determine if \(\sum_{n=1}^{\infty} a_n\) converges or diverges.

### Approach:

For the questions involving series (1, 2, and 3), we would typically use **comparison tests**, **ratio tests**, and known series for analysis. For question 4, we will work directly with the recursive formula to compute the initial terms and analyze their behavior.

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Would you like a detailed step-by-step solution for one of these questions? Let me know which one you'd like to tackle first!

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**Related Questions:**
1. How do we apply the ratio test to check convergence of a series?
2. What is a geometric series, and how does it relate to Problem 2?
3. How can we estimate the sum of an infinite series using finite partial sums?
4. What properties of a decreasing sequence ensure convergence of a series?
5. What are other common tests for series convergence?

**Tip:** The comparison test is often useful when dealing with complex series like in question 1, by comparing them to simpler, well-known convergent or divergent series.

Converge or diverge? Justify your answers.
1. a) ∑ (1 + 1/n)^2, b) ∑ (1 - 1/n)^2.

Let ∑ a_n be a series with positive terms. Prove relations for r_n decreasing/increasing. Use geometric series to find an upper bound for S_5 estimates. Find recursively defined terms a_2, a_3, a_4, and prove convergence/divergence.

Learn how to determine the convergence or divergence of infinite series, apply recursive formulas, and use geometric series for estimating sums. This problem set includes exercises on decreasing/increasing sequences and recursive definitions. Suitable for university-level students studying advanced calculus or mathematical series.

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