Monotonic and Bounded Sequences

The video tutorial covers monotonic and bounded sequences, explaining how sequences can be increasing or decreasing and how they can be bounded above or below. It provides examples to illustrate these concepts and discusses conditions for sequence convergence.

What is a monotonic sequence?

• A sequence that alternates between increasing and decreasing

• A sequence that remains constant

• A sequence that has no pattern

• A sequence that is always increasing or always decreasing

How can you prove a sequence is always increasing?

• By showing each term is a multiple of the previous term

• By showing each term is greater than or equal to the previous term

• By showing each term is equal to the previous term

• By showing each term is less than the previous term

In the example sequence a_n = 3n/(n+2), what is the nature of the sequence?

• It is not a monotonic sequence

• It is a constant sequence

• It is an increasing sequence

• It is a decreasing sequence

What does it mean for a sequence to be bounded above?

• The sequence has a maximum value it cannot exceed

• The sequence has a minimum value it cannot go below

• The sequence is always decreasing

• The sequence is always increasing

What is the conclusion if a sequence is both bounded and monotonic?

• The sequence diverges

• The sequence oscillates

• The sequence is constant

• The sequence converges

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

• Negative infinity

• One

• Zero

• Infinity

Is the sequence sine(n) monotonic?

• Yes, it is always increasing

• Yes, it is always decreasing

• No, it oscillates

• Yes, it is constant

What is the behavior of the sequence a_n = 3^n/n! as n increases?

• It remains constant

• It diverges to infinity

• It oscillates

• It converges to zero

How can you determine if a sequence is decreasing?

• By showing each term is greater than the next term

• By showing each term is a multiple of the next term

• By showing each term is less than the next term

• By showing each term is equal to the next term

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

• Negative one

• Infinity

• One

• Zero