14. Sequences & Series

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (–1)ⁿ (3n³ + 4n) / (6n³ + 5)

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = 8ⁿ / n!

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (–1)ⁿ / 0.9ⁿ

{Use of Tech} Repeated square roots

Consider the sequence defined by

aₙ₊₁ = √(2 + aₙ),a₀ = √2, for n = 0, 1, 2, 3, …

a.Evaluate the first four terms of the sequence {aₙ}.

State the exact values first, and then the approximate values.

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

a. Find the next two terms of the sequence.

{1, 2, 4, 8, 16, ......}

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

a. Find the next two terms of the sequence.

{1, 3, 9, 27, 81, ......}

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

a. Find the next two terms of the sequence.

{-5, 5, -5, 5, ......}

57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.

a. Find the first four terms of the sequence of heights {hₙ}.

h₀ = 20,r = 0.5

57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.

a. Find the first four terms of the sequence of heights {hₙ}.

h₀ = 30,r = 0.25

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

a.Write out the first five terms of the sequence.

Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

a.Write out the first five terms of the sequence.

Drug elimination

Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

For what values of r does the sequence {rⁿ} converge? Diverge?

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.

{1.00001ⁿ}