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Problem 10.2.17
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹(10n⁄(10n + 4))}
Problem 10.4.9
9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.
∑ (k = 0 to ∞) k / (2k + 1)
Problem 10.4.57
48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 1 to ∞) (5 / 6)⁻ᵏ
Problem 10.2.65
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{cos n / n}
Problem 10.5.53
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / k^(1 + p), p > 0
Problem 10.1.53
51–56. {Use of Tech} Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
aₙ₊₁ = 4aₙ + 1 a₀ = 1
Problem 10.6.63
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.)
Problem 10.5.11
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) (k² − 1) / (k³ + 4)
Problem 10.2.87
84–87. {Use of Tech} Sequences by recurrence relations
The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.
a. Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.
b. Use analytical methods to find the limit of the sequence.
{Use of Tech} aₙ₊₁ = √(2 + aₙ); a₀ = 3
Problem 10.4.35
23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) (k / (k + 10))ᵏ
Problem 10.2.55
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(−1)ⁿ / 2ⁿ}
Problem 10.1.49
49–50. Limits from graphs Consider the following sequences. Find the first four terms of the sequence .Based on part (a) and the figure, determine a plausible limit of the sequence.
aₙ = 2 + 2⁻ⁿ ; n = 1, 2, 3, …
Problem 10.8.3
1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 3 to ∞) (2k²) / (k² − k − 2)
Problem 10.2.77
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
{n¹⁰ / ln 20 n}
Problem 10.8.79
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) tan⁻¹(1 / √k)
Problem 10.2.41
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{ⁿ√(e³ⁿ⁺⁴)}
Problem 10.8.41
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2ᵏ / (3ᵏ − 2ᵏ)
Problem 10.4.33
23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) k / eᵏ
Problem 10.3.91
{Use of Tech} For what value of r does
∑ (k = 3 to ∞) r²ᵏ = 10?
Problem 10.6.5
Explain why the magnitude of the remainder in an alternating series (with terms that are nonincreasing in magnitude) is less than or equal to the magnitude of the first neglected term.
Problem 10.2.67
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{sin n / 2ⁿ}
Problem 10.3.23
21–42. Geometric series Evaluate each geometric series or state that it diverges.
23. ∑ (k = 0 to ∞) (–9/10)ᵏ
Problem 10.7.7
What test is advisable if a series involves a factorial term?
Problem 10.3.21
21–42. Geometric series Evaluate each geometric series or state that it diverges.
21. ∑ (k = 0 to ∞) (1/4)ᵏ
Problem 10.4.7
Define the remainder of an infinite series.
Problem 10.7.41
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (√k / k − 1)²ᵏ
Problem 10.8.67
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 1 to ∞) cot(–1 / j) / 2ʲ
Problem 10.7.11
9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) ((-1)ᵏ⁺¹) × ((10k³ + k) / (9k³ + k + 1))ᵏ
Problem 10.8.35
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2⁹k / kᵏ
Problem 10.6.51
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) cos(k) / k³
