14. Sequences & Series

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}

Compare the growth rates of {n¹⁰⁰} and {eⁿ⁄¹⁰⁰} as n → ∞.

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 }

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.

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a.The terms of the sequence {aₙ} increase in magnitude, so the limit of the sequence does not exist.

Sequences versus series

a.Find the limit of the sequence { (−⁴⁄₅)ᵏ }.

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 {0.000001aₙ} diverges.

Find two different explicit formulas for the sequence {1, -2, 3, -4, -5 .....}

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, 2, 4, 8, 16, ......}

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, 2, 4, 8, 16, ......}

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, ......}

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.

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

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

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, ......}

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