Understanding Sequences: Boundedness, Monotonicity, and Convergence

The video tutorial explains how to analyze a sequence given by a formula. It covers the simplification of the sequence, checks if it is bounded, and determines if it is monotonic. The tutorial also discusses whether the sequence converges or diverges, and if it converges, the value it converges to. The sequence is shown to be bounded, monotonic, and convergent, with a limit of zero.

What is the simplified form of the sequence a_n = n^4 / n^6?

• 1 / n^2

• n^2

• 1 / n^4

• n^6

What does it mean for a sequence to be bounded?

• It has a fixed number of terms.

• It is always decreasing.

• It has an upper and lower bound.

• It is always increasing.

What is the upper bound of the sequence 1 / n^2?

• Infinity

• 0

• 1

• n

How does the sequence 1 / n^2 behave as n increases?

• It increases without bound.

• It decreases and approaches zero.

• It oscillates between two values.

• It remains constant.

What characterizes a monotonic sequence?

• It has an upper and lower bound.

• It converges to a specific value.

• It is either always increasing or always decreasing.

• It has a fixed number of terms.

Is the sequence 1 / n^2 monotonic?

• Yes, it is always decreasing.

• No, it remains constant.

• No, it oscillates.

• Yes, it is always increasing.

What is the limit of the sequence 1 / n^2 as n approaches infinity?

• n

• 0

• 1

• Infinity

What does it mean for a sequence to converge?

• It is always increasing.

• It is always decreasing.

• It approaches a specific value as n increases.

• It has a fixed number of terms.

According to the theorem, what type of sequence will always converge?

• Any sequence with a limit.

• Any sequence that is bounded and monotonic.

• Any sequence that is unbounded.

• Any sequence that is not monotonic.

What is the lower bound of the sequence 1 / n^2?

• 0

• Infinity

• 1

• n