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Problem 10.4.65b
Loglog p-series Consider the series ∑ (k = 2 to ∞) 1 / (k(ln k)(ln ln k)ᵖ), where p is a real number.
b. Which of the following series converges faster? Explain.
∑ (k = 2 to ∞) 1 / (k(ln k)²) or ∑ (k = 3 to ∞) 1 / (k(ln k)(ln ln k)²)?
Problem 10.4.49
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 = 2 to ∞) 1 / eᵏ
Problem 10.4.47d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. Every partial sum Sₙ of the series ∑ (k = 1 to ∞) 1 / k² underestimates the exact value of ∑ (k = 1 to ∞) 1 / k².
Problem 10.4.47c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. Suppose f is a continuous, positive, decreasing function, for x ≥ 1, and aₖ = f(k), for k = 1, 2, 3, …. If ∑ (k = 1 to ∞) aₖ converges to L, then ∫ (1 to ∞) f(x) dx converges to L.
Problem 10.4.60
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 ∞) 3ᵏ / (k² + 1)
Problem 10.4.63
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 ∞) 3ᵏ⁺² / 5ᵏ
Problem 10.4.47a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The sum ∑ (k = 1 to ∞) 1 / 3ᵏ is a p-series.
Problem 10.4.47b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. The sum ∑ (k = 3 to ∞) 1 / √(k − 2) is a p-series.
Problem 10.4.41a
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
a. Find an upper bound for the remainder in terms of n.
41. ∑ (k = 1 to ∞) 1 / k⁶
Problem 10.4.41b
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.
41. ∑ (k = 1 to ∞) 1 / k⁶
Problem 10.4.41c
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.
41. ∑ (k = 1 to ∞) 1 / k⁶
Problem 10.4.41d
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.
41. ∑ (k = 1 to ∞) 1 / k⁶
Problem 10.4.43a
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
a. Find an upper bound for the remainder in terms of n.
43. ∑ (k = 1 to ∞) 1 / 3ᵏ
Problem 10.4.43b
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.
43. ∑ (k = 1 to ∞) 1 / 3ᵏ
Problem 10.4.43c
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.
43. ∑ (k = 1 to ∞) 1 / 3ᵏ
Problem 10.4.43d
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.
43. ∑ (k = 1 to ∞) 1 / 3ᵏ
Problem 10.4.39a
39–40. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, use Theorem 10.13 to complete the following.
a. Use Sₙ to estimate the sum of the series.
39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2
Problem 10.4.39b
39–40. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, use Theorem 10.13 to complete the following.
b. Find an upper bound for the remainder Rₙ.
39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2
Problem 10.4.39c
39–40. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, use Theorem 10.13 to complete the following.
c. Find lower and upper bounds (Lₙ and Uₙ, respectively) for the exact value of the series.
39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2
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 10.5.3
What comparison series would you use with the Comparison Test to determine whether
∑ (k = 1 to ∞) 1 / (k² + 1) converges?
Problem 10.5.5
What comparison series would you use with the Comparison Test to determine whether ∑ (k = 1 to ∞) 2ᵏ / (3ᵏ + 1) converges?
Problem 10.5.7
What comparison series would you use with the Limit Comparison Test to determine whether ∑ (k = 1 to ∞) (k² + k + 5) / (k³ + 3k + 1) converges?
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 10.5.17
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (−k)³ / (3k³ + 2)
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 10.5.15
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 4ᵏ / (5ᵏ − 3)
Problem 10.5.19
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 4 to ∞) (1 + cos²(k)) / (k − 3)
Problem 10.5.21
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) ((3k³ + 4)(7k² + 1)) / ((2k³ + 1)(4k³ − 1))
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