Which of the following sequences are unbounded?

Options

A. `(1+(1)/(n))^(n)`
B. `((2n+1)/(n+2))`
C. `(1+(1)/(n))^(n^(2))`
D. `tan n`

Explanation

To determine which of the given sequences are unbounded, we will analyze each sequence step by step.

### Step 1: Analyze the first sequence \( a_n = 1 + \frac{1}{n^n} \)

1. **Calculate the limit as \( n \) approaches infinity:**
\[ a_n = 1 + \frac{1}{n^n} \]
As \( n \) increases, \( n^n \) grows very large, making \( \frac{1}{n^n} \) approach 0.
\[ \lim_{n \to \infty} a_n = 1 + 0 = 1 \]
Since the limit exists and is finite, this sequence is **bounded**.

### Step 2: Analyze the second sequence \( a_n = \frac{2n + 1}{n + 2} \)

1. **Calculate the limit as \( n \) approaches infinity:**
\[ a_n = \frac{2n + 1}{n + 2} \]
Divide the numerator and denominator by \( n \):
\[ a_n = \frac{2 + \frac{1}{n}}{1 + \frac{2}{n}} \]
Taking the limit as \( n \) approaches infinity:
\[ \lim_{n \to \infty} a_n = \frac{2 + 0}{1 + 0} = 2 \]
Since the limit exists and is finite, this sequence is **bounded**.

### Step 3: Analyze the third sequence \( a_n = 1 + \frac{1}{n^{n^2}} \)

1. **Calculate the limit as \( n \) approaches infinity:**
\[ a_n = 1 + \frac{1}{n^{n^2}} \]
As \( n \) increases, \( n^{n^2} \) grows extremely large, making \( \frac{1}{n^{n^2}} \) approach 0.
\[ \lim_{n \to \infty} a_n = 1 + 0 = 1 \]
Since the limit exists and is finite, this sequence is **bounded**.

### Step 4: Analyze the fourth sequence \( a_n = 10n \)

1. **Calculate the limit as \( n \) approaches infinity:**
\[ a_n = 10n \]
As \( n \) increases, \( 10n \) goes to infinity.
\[ \lim_{n \to \infty} a_n = \infty \]
Since the limit is infinite, this sequence is **unbounded**.

### Conclusion

The sequences that are unbounded are:
- \( 10n \)

The sequences that are bounded are:
- \( 1 + \frac{1}{n^n} \)
- \( \frac{2n + 1}{n + 2} \)
- \( 1 + \frac{1}{n^{n^2}} \)

### Summary of Results
- **Bounded Sequences:**
- \( 1 + \frac{1}{n^n} \)
- \( \frac{2n + 1}{n + 2} \)
- \( 1 + \frac{1}{n^{n^2}} \)

- **Unbounded Sequences:**
- \( 10n \)