14. Sequences & Series

55–70. More sequences

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

aₙ = e⁻ⁿcosn

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.

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.

Write the first 6 terms of the sequence given by the recursive formula $a_{n}=a_{n-2}+a_{n-1}$ ; $a_1=1$ ; $a_2=1$.

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁. 

aₙ = 1/10ⁿ

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⁻ⁿif n > 5000 }

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

aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ

{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?

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

aₙ = 8ⁿ / n!

35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 

aₙ = 3 + cos(π*ⁿ) ; n = 1, 2, 3, …

Limits of sequences

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

{nsin(6 / n)}

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

c. Find an explicit formula for the nth term of the sequence.

{1, 3, 9, 27, 81, ......}

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

c. Find an explicit formula for the nth term of the sequence.

{-5, 5, -5, 5, ......}

55–70. More sequences

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

{nsin³(nπ / 2) / (n + 1)}"

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.

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 = 30,r = 0.25