14. Sequences & Series

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

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ · (k!) / (kᵏ)(Hint: Show that k! / kᵏ ≤ 2 / k², for k ≥ 3.)

Give an example (if possible) of a sequence {aₖ} that converges, while the series ∑ (from k = 1 to ∞) aₖ diverges.

Building a tunnel — first scenario

A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.

a.How far does the crew dig in 10 weeks? 20 weeks? N weeks?

Building a tunnel — first scenario

A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.

b.What is the longest tunnel the crew can build at this rate?

88–89. Binary numbers

Humans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbers—numbers consisting entirely of 0’s and 1’s. For this exercise, we consider binary numbers that have the form 0.b₁b₂b₃⋯, where each of the digits b₁, b₂, b₃, ⋯ is either 0 or 1. The base-10 representation of the binary number 0.b₁b₂b₃⋯ is the infinite series

b₁ / 2¹ + b₂ / 2² + b₃ / 2³ + ⋯

89. Computers can store only a finite number of digits and therefore numbers with nonterminating digits must be rounded or truncated before they can be used and stored by a computer.

a. Find the base-10 representation of the binary number 0.001̅1.

89–90. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, complete the following.

b. Find an upper bound for the remainder Rₙ.

89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

89–90. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, complete the following.

c. Find lower and upper bounds (Lₙ and Uₙ respectively) for the exact value of the series.

89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

Why does the value of a converging alternating series with terms that are nonincreasing in magnitude lie between any two consecutive terms of its sequence of partial sums?

8–32. {Use of Tech} Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − Sₙ| in using the nth partial sum Sₙ to estimate the value of the series S.

∑ (k = 1 to ∞) (−1)ᵏ / k⁴; n = 4

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.

ln 2 = ∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / k

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.

π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)