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₌₁2⁄[(2k − 1)(2k + 1)]

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

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, ......}

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

aₙ₊₁ = ⅓aₙ + 6;a₀ = 3.

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

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)?

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.

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.