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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 10.7.13
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 ∞) (k² / 4ᵏ)
Problem 10.2.5
For what values of r does the sequence {rⁿ} converge? Diverge?
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 10.3.97
Property of divergent series Prove Property 2 of Theorem 10.8: If ∑ aₖ diverges, then ∑ c aₖ also diverges, for any real number c ≠ 0.
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 10.3.79
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 2 to ∞) ln((k + 1)k⁻¹) / (ln k × ln(k + 1))
Problem 10.2.57
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
aₙ = (−1)ⁿ ⁿ√n
Problem 10.6.45
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ / k^(2/3)
Problem 10.3.33
21–42. Geometric series Evaluate each geometric series or state that it diverges.
33. ∑ (k = 4 to ∞) 1 / 5ᵏ
Problem 10.3.65
54–69. Telescoping series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.
65. ∑ (k = 1 to ∞) (1 / √(k + 1) – 1 / √(k + 3))
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 10.4.15
9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.
∑ (k = 2 to ∞) √k / (ln¹⁰ k)
Problem 10.5.27
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) (2 + (−1)ᵏ) / k²
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 10.5.9
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (k² + 4)
Problem 10.7.5
Simplify k! / (k + 2)! for any integer k ≥ 0.
Problem 10.8.61
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 1 / ln(eᵏ + 1)
Problem 10.5.23
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) sin(1 / k) / k²
Problem 10.7.3
Evaluate 1000!/998! without a calculator.
Problem 10.3.45
Periodic doses
Suppose you take 200 mg of an antibiotic every 6 hr. The half-life of the drug (the time it takes for half of the drug to be eliminated from your blood) is 6 hr. Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood. You may assume the steady-state amount is finite.
Problem 10.4.31
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 = 3 to ∞) 1 / (k − 2)⁴
Problem 10.2.81
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)
Problem 10.4.23
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^(–1/5)
Problem 10.6.25
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k¹/ᵏ
Problem 10.4.17
17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.
∑ (k = 1 to ∞) 1 / (∛(5k + 3))
Problem 10.8.51
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (7ᵏ + 11ᵏ) / 11ᵏ
Problem 10.8.83
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 2 to ∞) 1 / (j ln¹⁰j)
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