Understanding Sequences: Boundedness, Monotonicity, and Convergence

The video tutorial explores a sequence defined as a_n = (3/2)^n, examining its properties. It first determines if the sequence is bounded, concluding it is unbounded as it lacks an upper bound. Next, it checks if the sequence is monotonic, confirming it is monotonically increasing due to its geometric nature with a ratio greater than one. Finally, the tutorial analyzes convergence, concluding the sequence diverges as it increases without bound. The video provides a comprehensive understanding of the sequence's behavior through definitions, examples, and graphical analysis.

What is the common ratio of the sequence a_n = (3^n)/(2^n)?

• 1.5

• 0.5

• 2

• 3

Which of the following best describes a bounded sequence?

• A sequence with only an upper bound

• A sequence with only a lower bound

• A sequence with no bounds

• A sequence with both an upper and a lower bound

What is the lower bound of the sequence a_n = (3^n)/(2^n)?

• 0

• 1

• 2

• 3/2

How does the sequence a_n = (3^n)/(2^n) behave as n increases?

• It decreases without bound

• It remains constant

• It increases without bound

• It oscillates

What does it mean for a sequence to be monotonic?

• It has both an upper and a lower bound

• It is always increasing or always decreasing

• It oscillates between values

• It has a constant value

Is the sequence a_n = (3^n)/(2^n) monotonic?

• Yes, it is monotonically increasing

• Yes, it is monotonically decreasing

• No, it is not monotonic

• It oscillates

What is the formal method to determine if a sequence converges?

• Finding the common ratio

• Using limits as n approaches infinity

• Graphing the sequence

• Checking if it has an upper bound

Does the sequence a_n = (3^n)/(2^n) converge or diverge?

• It oscillates

• It diverges

• It converges to a finite value

• It converges to zero

Why does the sequence a_n = (3^n)/(2^n) diverge?

• The sequence has an upper bound

• The denominator increases faster than the numerator

• The numerator increases faster than the denominator

• Both numerator and denominator increase at the same rate

Which of the following properties does the sequence a_n = (3^n)/(2^n) NOT have?

• It diverges

• It converges

• It is monotonic

• It is unbounded