Let's analyze and solve the problems in the image related to sequences:

Problem 2: Determine if the sequence is increasing, decreasing, or not monotonic.

(a) $\left\{\frac{2n-1}{4n-1}\right\}$

We need to check whether the sequence is increasing or decreasing by finding the difference between consecutive terms or evaluating the derivative (for large $n$).

• For large $n$, $\frac{2n-1}{4n-1}$ behaves approximately like $\frac{2}{4} = \frac{1}{2}$, so the sequence seems to stabilize.
• To formally check if the sequence is monotonic, we can evaluate the first difference: $a_{n+1} - a_n = \frac{2(n+1)-1}{4(n+1)-1} - \frac{2n-1}{4n-1}$ This can be simplified to check if the result is positive (increasing) or negative (decreasing).

(b) $\left\{3 - 2ne^{-n}\right\}$

We observe the general behavior of the term $2ne^{-n}$:

• As $n \to \infty$, $e^{-n}$ approaches 0 very quickly, so $2ne^{-n} \to 0$.
• The sequence approaches $3$ as $n$ increases. This suggests the sequence is increasing because $2ne^{-n}$ decreases with larger $n$.

(c) $\left\{\frac{n^n}{n!}\right\}$

This sequence grows very rapidly because $n^n$ increases much faster than $n!$. To determine if it is increasing or decreasing, we could use Stirling's approximation for large $n$ to approximate both $n^n$ and $n!$ and check the behavior. This sequence grows extremely fast, indicating it is likely increasing for large $n$.

Problem 3: Determine if the sequence is bounded.

(a) $\left\{\frac{1 - n}{2 + n}\right\}$

• For large $n$, the sequence behaves like $\frac{-n}{n} = -1$.
• The sequence approaches $-1$ as $n \to \infty$, but for small $n$, the values will vary.
• The sequence is bounded because it does not tend to infinity or negative infinity; it is limited by $-1$.

(b) $\left\{2 + \frac{(-1)^n}{n}\right\}$

• As $n \to \infty$, the term $\frac{(-1)^n}{n}$ tends to 0.
• The sequence oscillates between slightly above and slightly below 2, so it is bounded between $2 - \frac{1}{n}$ and $2 + \frac{1}{n}$.
• Therefore, the sequence is bounded.

Would you like further details or explanations on any of these sequences?

Here are some follow-up questions to deepen understanding:

1. How can Stirling's approximation be used to simplify large factorials in sequences?
2. What is the difference between bounded and monotonic sequences?
3. How do you prove rigorously whether a sequence is increasing or decreasing?
4. What are the conditions for a sequence to be unbounded?
5. How do factorial and exponential terms in sequences affect their growth?

Tip: When analyzing sequences for monotonicity, sometimes it's easier to compare the $n+1$-th term with the $n$-th term directly.