Explain why or why not

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

e. The sequence aₙ = n² / (n² + 1) converge.

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞) k√k / k³

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞) k⁵ e⁻ᵏ

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞) k! / (eᵏ kᵏ)

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞) 2ᵏ / eᵏ

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.

∑ (from k = 0 to ∞) ((1/3)ᵏ + (4/3)ᵏ)

89–90. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, complete the following.

b. Find an upper bound for the remainder Rₙ.

89. ∑ (from k = 1 to ∞) 1 / k⁵ ; n = 5

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) (−1)ᵏ⁺¹ / k³⁄⁷

a. Does the sequence { k/(k + 1) } converge? Why or why not?

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

aₙ = (–1)ⁿ / 0.9ⁿ

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)

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) (−2)ᵏ⁺¹ / k²

Explain why or why not

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

a. The terms of the sequence {aₙ} increase in magnitude, so the limit of the sequence does not exist.

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.

41. ∑ (k = 1 to ∞) 1 / k⁶

{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.

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

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

87. Explain why or why not

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

a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ 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.

a. Let dₙ equal the amount of medication (in mg) in the bloodstream after n doses, where d₁ = 75.

Find a recurrence relation for dₙ.

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, ......}

{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.

a. Write the first five terms of the sequence {Bₙ}.

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, ......}

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

Find the first term a and the ratio r of each geometric series.

a. ∑ k = 0 to ∞ (2/3) × (1/5)ᵏ

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

a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.

∑ (k = 0 to ∞) (–2/7)ᵏ

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.

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, ......}

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

a. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.

88–89. Binary numbers

Humans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbers—numbers consisting entirely of 0’s and 1’s. For this exercise, we consider binary numbers that have the form 0.b₁b₂b₃⋯, where each of the digits b₁, b₂, b₃, ⋯ is either 0 or 1. The base-10 representation of the binary number 0.b₁b₂b₃⋯ is the infinite series

b₁ / 2¹ + b₂ / 2² + b₃ / 2³ + ⋯

89. Computers can store only a finite number of digits and therefore numbers with nonterminating digits must be rounded or truncated before they can be used and stored by a computer.

a. Find the base-10 representation of the binary number 0.001̅1.

71. Evaluating an infinite series two ways

Evaluate the series

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

a. Use a telescoping series argument.

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)]