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Problem 10.2.5
For what values of r does the sequence {rⁿ} converge? Diverge?
Problem 10.2.7
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}
Problem 10.2.9
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.
{(−0.7)ⁿ}
Problem 10.2.11
Compare the growth rates of {n¹⁰⁰} and {eⁿ⁄¹⁰⁰} as n → ∞.
Problem 10.2.49
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{bₙ}, where
bₙ = { n / (n + 1) if n ≤ 5000
ne⁻ⁿ if n > 5000 }
Problem 10.2.83b
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.
Problem 10.2.83c
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.
Problem 10.2.83f
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
f. If the sequence {aₙ} diverges, then the sequence {0.000001 aₙ} diverges.
Problem 10.2.93b
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.
Problem 10.2.71a
{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.
Problem 10.2.71b
{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?
Problem 10.2.71c
{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.
Problem 10.2.73b
{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}.
Problem 10.2.73c
{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?
Problem 10.2.83d
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.
Problem 10.2.85
84–87. {Use of Tech} Sequences by recurrence relations
The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.
a. Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.
b. Use analytical methods to find the limit of the sequence.
aₙ₊₁ = 2aₙ (1 − aₙ); a₀ = 0.3
Problem 10.2.87
84–87. {Use of Tech} Sequences by recurrence relations
The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.
a. Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.
b. Use analytical methods to find the limit of the sequence.
{Use of Tech} aₙ₊₁ = √(2 + aₙ); a₀ = 3
Problem 10.2.89a
{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.
Problem 10.2.89c
{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?
Problem 10.2.97b
{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?
Problem 10.3.19a
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
Problem 10.3.85
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 1 to ∞) (((1/6)ᵏ + (1/3)ᵏ) × k⁻¹)
Problem 10.3.79
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 2 to ∞) ln((k + 1)k⁻¹) / (ln k × ln(k + 1))
Problem 10.3.71b
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.
Problem 10.3.83
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 1 to ∞) ((1/3) × (5/6)ᵏ + (3/5) × (7/9)ᵏ)
Problem 10.3.47
46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).
47. 0.3̅ = 0.333…
Problem 10.3.71a
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.
Problem 10.3.53
46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).
53. 0.00952̅ = 0.00952952…
Problem 10.3.75
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)
Problem 10.3.5a
Find the first term a and the ratio r of each geometric series.
a. ∑ k = 0 to ∞ (2/3) × (1/5)ᵏ