Understanding Sequences: Boundedness, Monotonicity, and Convergence

The video tutorial explores the sequence a_n = (4^n)/(5^n), examining its properties. It discusses whether the sequence is bounded or unbounded, monotonic or not, and whether it converges or diverges. The sequence is identified as a geometric sequence with a common ratio less than one, making it bounded and monotonic. The tutorial concludes that the sequence converges to zero, supported by a theorem on bounded and monotonic sequences.

What is the initial form of the sequence a_n discussed in the video?

• a_n = (4n)/(5n)

• a_n = (5^n)/(4^n)

• a_n = (n^4)/(n^5)

• a_n = (4^n)/(5^n)

What does it mean for a sequence to be bounded?

• It has no bounds.

• It has both an upper and a lower bound.

• It has only a lower bound.

• It has only an upper bound.

What are the bounds of the sequence a_n = (4^n)/(5^n)?

• Bounded above by 1 and below by 1/5

• Bounded above by 1 and below by 0

• Bounded above by 4/5 and below by 0

• Bounded above by 5/4 and below by 1

What characterizes a monotonically decreasing sequence?

• Each term is less than or equal to the previous term.

• Each term is greater than the previous term.

• Each term is greater than or equal to the previous term.

• Each term is equal to the previous term.

Why is the sequence a_n = (4^n)/(5^n) considered monotonic?

• It alternates between increasing and decreasing.

• It is decreasing with each term.

• It remains constant.

• It is increasing with each term.

What theorem is used to determine the convergence of the sequence?

• A sequence that is bounded and monotonic will diverge.

• A sequence that is bounded and monotonic will converge.

• A sequence that is unbounded and non-monotonic will diverge.

• A sequence that is unbounded and monotonic will converge.

To what value does the sequence a_n = (4^n)/(5^n) converge?

• Infinity

• 0

• 4/5

• 1

What happens to the fraction (4^n)/(5^n) as n approaches infinity?

• It approaches 1.

• It approaches infinity.

• It approaches 4/5.

• It approaches 0.

What is the significance of the limit of a sequence?

• It determines if the sequence converges or diverges.

• It determines if the sequence is monotonic.

• It determines the first term of the sequence.

• It determines if the sequence is bounded.

What are the three main properties of the sequence discussed in the video?

• Unbounded, non-monotonic, and diverges

• Unbounded, monotonic, and converges

• Bounded, non-monotonic, and diverges

• Bounded, monotonic, and converges