{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.

67–70. Formulas for sequences of partial sums Consider the following infinite series.

a. Find the first four partial sums S₁, S₂, S₃, S₄ of the series.

∑⁽∞⁾ₖ₌₁ 2⁄[(2k − 1)(2k + 1)]

{Use of Tech} Periodic dosing

Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.

a. Find a recurrence relation for the sequence {dₙ} that gives the amount of drug in the blood after the nᵗʰ dose, where d₁ = 80.

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

a. Find an upper bound for the remainder in terms of n.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

39–40. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, use Theorem 10.13 to complete the following.

a. Use Sₙ to estimate the sum of the series.

39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

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

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series.

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a. If limₙ→∞ aₙ = 1 and limₙ→∞ bₙ = 3, then limₙ→∞ (bₙ / aₙ) = 3.

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a. The sequence of partial sums for the series 1 + 2 + 3 + ⋯ is {1, 3, 6, 10, …}.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A series that converges must converge absolutely.

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.

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

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

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

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.

{Use of Tech} Fibonacci sequence

The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about A.D. 1200 as a model for the growth of rabbit populations.

It is given by the recurrence relation: fₙ₊₁ = fₙ + fₙ₋₁, for n = 1, 2, 3, … where f₀ = 1 and f₁ = 1. Each term of the sequence is the sum of its two predecessors. 

b. Is the sequence bounded?

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.

41. ∑ (k = 1 to ∞) 1 / k⁶

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

b. Find an explicit formula for the 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.

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.

b. Find an explicit formula for the nth term of the sequence {hₙ}.

h₀ = 30, r = 0.25

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

b. Find an explicit formula for the 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.

Suppose the sequence {aₙ}⁽∞⁾ₙ₌₀ is defined by the recurrence relation

aₙ₊₁ = ⅓ aₙ + 6; a₀ = 3.

b. Explain why {aₙ}⁽∞⁾ₙ₌₀ converges and find the limit.

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

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

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

Loglog p-series Consider the series ∑ (k = 2 to ∞) 1 / (k(ln k)(ln ln k)ᵖ), where p is a real number.

b. Which of the following series converges faster? Explain.

∑ (k = 2 to ∞) 1 / (k(ln k)²) or ∑ (k = 3 to ∞) 1 / (k(ln k)(ln ln k)²)?

18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.

b. Evaluate the series using Theorem 10.7.

∑ (k = 0 to ∞) (–2/7)ᵏ

67–70. Formulas for sequences of partial sums Consider the following infinite series.

b. Find a formula for the nth partial sum Sₙ of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, S₈ of the infinite series.

∑⁽∞⁾ₖ₌₁ 2⁄[(2k − 1)(2k + 1)]

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. A series that converges absolutely must converge.

{Use of Tech} A savings plan

James begins a savings plan in which he deposits $100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.

To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits $100.

Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = $0.

b. Find a recurrence relation that generates the sequence {Bₙ}.

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

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

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

71. Evaluating an infinite series two ways

Evaluate the series

∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.

b. Use a geometric series argument with Theorem 10.8.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. The sum ∑ (k = 3 to ∞) 1 / √(k − 2) is a p-series.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. (2n)! / (2n − 1)! = 2n