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Problem 10.R.17
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = 8ⁿ / n!
Problem 10.R.31
27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 1 to ∞) ln((2k + 1) / (2k − 1))
Problem 10.R.23
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (–1)ⁿ / 0.9ⁿ
Problem 10.R.41
Finding steady states using infinite series Solve Exercise 40 by expressing the amount of aspirin in your blood as a geometric series and evaluating the series.
Problem 10.R.11b
b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?
Problem 10.R.9a
Sequences versus series
a. Find the limit of the sequence { (−⁴⁄₅)ᵏ }.
Problem 10.R.19
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)
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.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.15
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ
Problem 10.R.49
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞) k⁴ / √(9k¹² + 2)
Problem 10.R.63
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞) 3 / (2 + eᵏ)
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.71
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞) (1 − cos(1 / k))²
Problem 10.R.57
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞) 5ᵏ / 2²ᵏ⁺¹
Problem 10.R.3
Geometric sums
Evaluate the geometric sums
∑ (from k = 0 to 9) (0.2)ᵏ and ∑ (from k = 2 to 9) (0.2)ᵏ.
Problem 10.R.1g
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
g. The series ∑ (from k = 1 to ∞) (k² / (k² + 1)) converges.
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.R.1a
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.
Problem 10.R.47
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞) (7 + sin k) / k²
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.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.37
27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 1 to ∞) 2ᵏ / 3ᵏ⁺²
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.75
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞) tanh(k)
Problem 10.R.39
Express 0.314141414… as a ratio of two integers.
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.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.1e
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
e. The sequence aₙ = n² / (n² + 1) converge.
Problem 10.R.65
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞) k√k / k³