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Problem 10.R.77
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ⁺¹ / k³⁄⁷
Problem 10.R.81
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ⁺¹(k² + 4) / (2k² + 1)
Problem 10.R.83
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏk·e⁻ᵏ
Problem 10.R.87
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−2)ᵏ⁺¹ / k²
Problem 10.R.1c
Determine whether the following statements are true and give an explanation or counterexample.
c. The terms of the sequence of partial sums of the series ∑ aₖ approach 5/2, so the infinite series converges to 5/2.
Problem 10.R.25a
25–26. Recursively defined sequences
The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.
a. Find the first five terms a₀, a₁, ..., a₄ of each sequence.
25. aₙ₊₁ = (1 / 2) aₙ + 8; a₀ = 80
Problem 10.R.25b
25–26. Recursively defined sequences
The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.
b. Determine the limit of each sequence.
25. aₙ₊₁ = (1 / 2) aₙ + 8; a₀ = 80
Problem 10.R.7
Give an example (if possible) of a sequence {aₖ} that converges, while the series ∑ (from k = 1 to ∞) aₖ diverges.
Problem 10.R.95a
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?
Problem 10.R.95b
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?
Problem 10.R.89b
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
Problem 10.R.89c
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
Problem 10.R.93
{Use of Tech} Error in a finite alternating sum
How many terms of the series ∑ (from k = 1 to ∞) (−1)ᵏ⁺¹ / k⁴ must be summed to ensure that the approximation error is less than 10⁻⁸?
Problem 10.R.91
Estimate the value of the series ∑ (from k = 1 to ∞) 1 / (2k + 5)³ to within 10⁻⁴ of its exact value.
Problem 10.1.67b
67–70. Formulas for sequences of partial sums Consider the following infinite series.
b. Find a formula for the nth partial sum Sₙ of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, S₈ of the infinite series.
∑⁽∞⁾ₖ₌₁ 2⁄[(2k − 1)(2k + 1)]
Problem 10.1.7
The first ten terms of the sequence {(1 + 1/10ⁿ)^10ⁿ}∞ ₙ₌₁ are rounded to 8 digits right of the decimal point (see table). Make a conjecture about the limit of the sequence.
n an
1 2.59374246
2 2.70481383
3 2.71692393
4 2.71814593
5 2.71826824
6 2.71828047
7 2.71828169
8 2.71828179
9 2.71828204
10 2.71828203
Problem 10.1.2
Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.
Problem 10.1.3
Suppose the sequence { aₙ} is defined by the recurrence relation a₍ₙ₊₁₎ = n · aₙ , for n=1, 2, 3 ...., where a₁ = 1. Write out the first five terms of the sequence.
Problem 10.1.9
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.
Problem 10.1.10
Given the series ∑∞ₖ₌₁ k, evaluate the first four terms of its sequence of partial sums Sₙ = ∑ⁿₖ₌₁ k.
Problem 10.1.13
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = 1/10ⁿ
Problem 10.1.15
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (−1)ⁿ / 2ⁿ
Problem 10.1.17
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)
Problem 10.1.19
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = 1 + sin(πn / 2)
Problem 10.1.21
21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 2aₙ; a₁ = 2
Problem 10.1.23
"21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 3aₙ-12; a₁ = 10
Problem 10.1.37
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 = 1⁄10ⁿ ; n = 1, 2, 3, …
Problem 10.1.39
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, …
Problem 10.1.43
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.
{Use of Tech} aₙ₊₁ = (aₙ⁄₁₁ )+ 50 ; a₀ = 50
Problem 10.1.63
61–66. Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.
4 + 0.9 + 0.09 + 0.009 + ⋯