14. Sequences & Series

Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.

{n¹⁰ / ln20n}

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.

13–52. Limits of sequences

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

{ⁿ√(e³ⁿ⁺⁴)} 

13–52. Limits of sequences

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

{√((1 + 1 / 2n)ⁿ)}

Finding Terms of a Sequence

Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.

aₙ = 1 / n!

Write a recursive formula for the arithmetic sequence.

$\(\left\]\lbrace\)8,2,-4,-10,\(\ldots\[\right\]\rbrace\)$

55–70. More sequences

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

{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}

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.

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.

55–70. More sequences

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

{(−1)ⁿ / 2ⁿ}

The terms of a sequence of partial sums are defined by Sₙ = ∑ⁿₖ₌₁ k² , for n=1, 2, 3, .....Evaluate the first four terms of the sequence.

In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.

aₙ = 2ⁿ 3ⁿ / n!

13–52. Limits of sequences

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

{(3n³ − 1)⁄(2n³ + 1)}

{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?

Find the general formula for the arithmetic sequence below. Without using a recursive formula, calculate the $30^{\(\th\)}$ term.

$\(\left\]\lbrace\)-9,-4,1,6,\(\ldots\[\right\]\rbrace\)$

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