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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 \(n\).

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### 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.

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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.

2. Determine if the sequence is increasing, decreasing, or not monotonic.
(a) { (2n - 1) / (4n - 1) }
(b) { 3 - 2ne^{-n} }
(c) { n^n / n! }

3. Determine if the sequence is bounded.
(a) { (1 - n) / (2 + n) }
(b) { 2 + (-1)^n / n }

Analyze sequences for monotonicity and boundedness, including detailed solutions for sequences like (2n - 1)/(4n - 1), 3 - 2ne^{-n}, and n^n/n!. Learn how to determine whether sequences are increasing, decreasing, or bounded, suitable for undergraduate math students.

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