Give an example of a bounded sequence that has a limit. Choose the correct answer below. A. $$ a_{n}=(-1)^{(n+1)}, n \geq 1 $$ B. $$ a_{n}= an n, n \geq 1 $$ C. $$ a_{n}=\frac{e^{n}}{n}, n \geq 1 $$ D. $$ a_{n}=\frac{n}{n+1}, n \geq 1 $$

Question

Give an example of a bounded sequence that has a limit. Choose the correct answer below. A. $$ a_{n}=(-1)^{(n+1)}, n \geq 1 $$ B. $$ a_{n}=	an n, n \geq 1 $$ C. $$ a_{n}=\frac{e^{n}}{n}, n \geq 1 $$ D. $$ a_{n}=\frac{n}{n+1}, n \geq 1 $$

UpStudy AI · Accepted Answer

To determine which of the given sequences is bounded and has a limit, let's analyze each option step by step.

### Option A: $$ a_{n}=(-1)^{(n+1)}, n \geq 1 $$
- This sequence alternates between 1 and -1:
  - $$ a_1 = 1 $$
  - $$ a_2 = -1 $$
  - $$ a_3 = 1 $$
  - $$ a_4 = -1 $$
- The sequence is bounded (between -1 and 1), but it does not converge to a single limit. Therefore, this option does not satisfy the conditions.

### Option B: $$ a_{n}=\tan n, n \geq 1 $$
- The tangent function is unbounded as $$ n $$ increases, since it approaches infinity at odd multiples of $$ \frac{\pi}{2} $$. Thus, this sequence is not bounded and does not have a limit.

### Option C: $$ a_{n}=\frac{e^{n}}{n}, n \geq 1 $$
- As $$ n $$ increases, $$ e^n $$ grows much faster than $$ n $$, leading to:
  $$
  \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{e^n}{n} = \infty
  $$
- This sequence is unbounded and does not have a finite limit.

### Option D: $$ a_{n}=\frac{n}{n+1}, n \geq 1 $$
- This sequence can be simplified:
  $$
  a_n = \frac{n}{n+1} = 1 - \frac{1}{n+1}
  $$
- As $$ n $$ approaches infinity, $$ a_n $$ approaches:
  $$
  \lim_{n \to \infty} a_n = 1 - 0 = 1
  $$
- The sequence is bounded (between 0 and 1) and converges to 1.

### Conclusion
The correct answer is:
**D. $$ a_{n}=\frac{n}{n+1}, n \geq 1 $$**.

This sequence is bounded and has a limit of 1.
The correct answer is D: $$ a_{n}=\frac{n}{n+1}, n \geq 1 $$. This sequence is bounded and converges to 1.

Give an example of a bounded sequence that has a limit. Choose the correct answer below. A. \( a_{n}=(-1)^{(n+1)}, n \geq 1 \) B. \( a_{n}=\tan n, n \geq 1 \) C. \( a_{n}=\frac{e^{n}}{n}, n \geq 1 \) D. \( a_{n}=\frac{n}{n+1}, n \geq 1 \)

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