Explain why or why not

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

b. If a sequence of positive numbers converges, then the sequence is decreasing.

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.

b. Find an upper bound for the remainder Rₙ.

39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

Explain why or why not

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

b. If limₙ→∞ aₙ = 0 and limₙ→∞ bₙ = ∞, then limₙ→∞ aₙbₙ = 0.

{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.

b. Use a calculator to estimate this limit. In the long run, how much drug is in the person’s blood?

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 = 20, r = 0.5

{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.

c. Assuming the sequence has a limit, confirm the result of part (b) by finding the limit of {dₙ} directly.

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

c. Find an explicit formula for the nth term of the sequence.

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

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

c. Find an explicit formula for the nth term of the sequence.

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

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

c. Make a conjecture for the value of the series.

∑⁽∞⁾ₖ₌₁ 2⁄[(2k − 1)(2k + 1)]

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

c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

{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.

c. How many months are needed to reach a balance of $5000?

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.

c. Find lower and upper bounds (Lₙ and Uₙ, respectively) for the exact value of the series.

39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

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

c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.

41. ∑ (k = 1 to ∞) 1 / k⁶

Explain why or why not

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

c. The convergent sequences {aₙ} and {bₙ} differ in their first 100 terms, but aₙ = bₙ for n > 100.

It follows that limₙ→∞ aₙ = limₙ→∞ bₙ.

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

c. Find a recurrence relation that generates 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.

Explain why or why not

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

c. If the terms of the sequence {aₙ} are positive and increasing, then the sequence of partial sums for the series ∑⁽∞⁾ₖ₌₁ aₖ diverges.

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

c. Find an explicit formula for the nth term of the sequence.

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

87. Explain why or why not

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

c. If ∑ aₖ converges, then ∑ (aₖ + 0.0001) converges.

{Use of Tech} Drug Dosing

A patient takes 75 mg of a medication every 12 hours; 60% of the medication in the blood is eliminated every 12 hours.

c. Find the limit of the sequence. What is the physical meaning of this limit?

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

c. Find a recurrence relation that generates 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.

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

c. Suppose f is a continuous, positive, decreasing function, for x ≥ 1, and aₖ = f(k), for k = 1, 2, 3, …. If ∑ (k = 1 to ∞) aₖ converges to L, then ∫ (1 to ∞) f(x) dx converges to L.

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

c. A series that converges conditionally must converge.

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

c. If lim (as k → ∞) ᵏ√|aₖ| = 1/4, then ∑ 10aₖ converges absolutely.

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

d. If ∑ aₖ diverges, then ∑ |aₖ| diverges.

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

d. The Ratio Test is always inconclusive when applied to ∑ aₖ, where aₖ is a nonzero rational function of k.

Explain why or why not

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

d. If {aₙ} = {1, ½, ⅓, ¼, ⅕, …} and

{bₙ} = {1, 0, ½, 0, ⅓, 0, ¼, 0, …},

then limₙ→∞ aₙ = limₙ→∞ bₙ.

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.

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.

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

d. Every partial sum Sₙ of the series ∑ (k = 1 to ∞) 1 / k² underestimates the exact value of ∑ (k = 1 to ∞) 1 / k².

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

d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.

41. ∑ (k = 1 to ∞) 1 / k⁶

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

d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ