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.

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!

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.