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Problem 10.5.31
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 20 / (∛k + √k)
Problem 10.5.25
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (2k − √k)
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 10.5.50
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 2 to ∞) (5 ln k) / k
Problem 10.5.41
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) (1 + 2 / k)ᵏ
Problem 10.5.45
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 3 to ∞) 1 / ln k
Problem 10.5.57
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 2 to ∞) 1 / (k ln k)
Problem 10.5.59
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) tan (1 / k)
Problem 10.5.51
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) k⁸ / (k¹¹ + 3)
Problem 10.5.37a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.
Problem 10.5.37d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. When applying the Limit Comparison Test, an appropriate comparison series for ∑ (k = 1 to ∞) (k² + 2k + 1) / (k⁵ + 5k + 7) is ∑ (k = 1 to ∞) 1 / k³.
Problem 10.5.35
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
{Use of Tech} ∑ (k = 1 to ∞) 1 / (4 ln k)
Problem 10.5.63
Series of squares Prove that if ∑ aₖ is a convergent series of positive terms, then the series ∑ aₖ² also converges.
Problem 10.5.39
38–39. Examining a series two ways Determine whether the following series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer.
39. ∑ (k = 1 to ∞) 1 / (k² + 2k + 1)
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 10.6.11
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)
Problem 10.6.13
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)
Problem 10.6.17
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)
Problem 10.6.21
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 2 to ∞) (−1)ᵏ (1 + 1/k)
Problem 10.6.23
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ (k¹¹ + 2k⁵ + 1) / [4k(k¹⁰ + 1)]
Problem 10.6.25
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k¹/ᵏ
Problem 10.6.65a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. A series that converges must converge absolutely.
Problem 10.6.65b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. A series that converges absolutely must converge.
Problem 10.6.65c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. A series that converges conditionally must converge.
Problem 10.6.65d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. If ∑ aₖ diverges, then ∑ |aₖ| diverges.
Problem 10.6.7
Is it possible for a series of positive terms to converge conditionally? Explain.
Problem 10.6.9
Is it possible for an alternating series to converge absolutely but not conditionally?
Problem 10.6.59
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ · tan⁻¹(k) / k³
Problem 10.6.61
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)
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.)