14. Sequences & Series

Finding a Sequence’s Formula

In Exercises 13–30, find a formula for the nth term of the sequence.

1, -4, 9, -16, 25, …Squares of the positive integers, with alternating signs

{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?

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 (3n³ + 4n) / (6n³ + 5)

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.

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

Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.

{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.

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.

{cosn / n}

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.

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

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{n³⁄(n⁴ + 1)}  

51–56. {Use of Tech} Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.

aₙ₊₁ = 4aₙ + 1 a₀ = 1

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

49–50. Limits from graphs Consider the following sequences. Find the first four terms of the sequence .Based on part (a) and the figure, determine a plausible limit of the sequence.

aₙ = 2 + 2⁻ⁿ;n = 1, 2, 3, …