Understanding Sequences: Boundedness, Monotonicity, and Convergence

The video tutorial explores a sequence defined as a_n = n^4 / n^3, simplifying it to a_n = n. It examines whether the sequence is bounded, monotonic, and if it converges or diverges. The sequence is found to be unbounded, monotonically increasing, and divergent, as it increases without bound.

What is the simplified form of the sequence a_n = n^4 / n^3?

• a_n = n^2

• a_n = n

• a_n = 1/n

• a_n = n^3

What does it mean for a sequence to be bounded?

• It has a fixed number of terms.

• It has both an upper and a lower bound.

• It is always increasing.

• It converges to a specific value.

What is the lower bound of the sequence a_n = n?

• n

• 1

• Infinity

• 0

Why is the sequence a_n = n considered unbounded?

• It decreases without limit.

• It has both upper and lower bounds.

• It converges to zero.

• It has a lower bound but no upper bound.

What characterizes a monotonic sequence?

• It has a fixed number of terms.

• It is always increasing or always decreasing.

• It has both an upper and a lower bound.

• It converges to a specific value.

Is the sequence a_n = n monotonically increasing or decreasing?

• Neither

• Both

• Monotonically decreasing

• Monotonically increasing

What is the definition of a convergent sequence?

• A sequence that approaches a specific limit.

• A sequence that has both an upper and a lower bound.

• A sequence that has a fixed number of terms.

• A sequence that is always increasing.

Does the sequence a_n = n converge or diverge?

• Converges to a finite value

• Diverges

• Converges to infinity

• Converges to zero

What happens to the terms of the sequence a_n = n as n approaches infinity?

• They approach zero.

• They remain constant.

• They increase without bound.

• They decrease without bound.

Why does the sequence a_n = n not have a limit?

• Because it is not monotonic.

• Because it is bounded.

• Because it increases without bound.

• Because it decreases without bound.