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Consider the sequence $\displaystyle{\lbrace a_n \rbrace = \left\lbrace {\frac{\left(-1\right)^{n}\cdot 3n}{n+1}} \right\rbrace }$ . Graph this sequence and use your graph to help you answer the following questions.

Part 1: Is the sequence bounded?

Is the sequence $\lbrace a_n \rbrace$ bounded above by a function? If it is, enter the function of the variable $n$ that provides the “best” and “most obvious” upper bound; otherwise, enter DNE for does not exist.

What is the limit of the function from part (a) as $n \to \infty$ ? Enter a number, or enter DNE.

Is the sequence $\lbrace a_n \rbrace$ bounded below by a function? If it is, enter the function of the variable $n$ that provides the “best” and “most obvious” lower bound; otherwise, enter DNE for does not exist.

What is the limit of the function from part (c) as $n \to \infty$ ? Enter a number, or enter DNE.

Is the sequence $\lbrace a_n \rbrace$ bounded above by a number? Enter a number or enter DNE.

Is the sequence $\lbrace a_n \rbrace$ bounded below by a number? Enter a number or enter DNE.

Select all that apply: The sequence $\lbrace a_n \rbrace$ is
A. bounded below.
B. bounded above.
C. unbounded.
D. bounded.

Part 2: Is the sequence monotonic?

The sequence $\lbrace a_n \rbrace$ is

A. increasing.
B. alternating
C. decreasing.
D. none of the above

Part 3: Does the sequence converge?

The sequence $\lbrace a_n \rbrace$ is .

The limit of the sequence $\lbrace a_n \rbrace$ is . Enter a number or DNE.

Part 4: Conceptual follow up questions

When you first look at the sequence $\displaystyle \left\lbrace \frac{\left(-1\right)^{n}\cdot 3n}{n+1} \right\rbrace$ , you expect it to
A. converge to both $-3$ and $3$ because the odd index terms tend to $-3$ as $n \to \infty$ , while the even index terms tend to $3$ as $n \to \infty$ .
B. diverge because the odd index terms tend to $-3$ as $n \to \infty$ , while the even index terms tend to $3$ as $n \to \infty$ , so there is not one single value for the limit of the sequence.
C. diverge because alternating sequences always diverge.

When you first look at the sequence $\displaystyle \left\lbrace \frac{\left(-1\right)^{n}\cdot 3}{n+1} \right\rbrace$ , you expect it to
A. diverge because alternating sequences always diverge.
B. diverge because the odd index terms tend to $-3$ as $n \to \infty$ , while the even index terms tend to $3$ as $n \to \infty$ , so there is not one single value for the limit of the sequence.
C. converge to $0$ because $\frac{3}{n+1} \leq \frac{\left(-1\right)^{n}\cdot 3}{n+1} \leq \frac{3}{n+1}$ and both $\frac{-3}{n+1}$ and $\frac{3}{n+1}$ converge to $0$ as $n \to \infty$ .

If a sequence is alternating, it converge.