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Problem 10.8.65
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (cos(1 / k) – cos(1 / (k + 1)))
Problem 10.2.69
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(75 n⁻¹ / 99ⁿ) + (5ⁿ sin n / 8ⁿ)}
Problem 10.7.17
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 ∞) ((-7)ᵏ / k²)
Problem 10.7.29
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 × k²ᵏ) / (k! × k!)
Problem 10.1.37
35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why.
aₙ = 1⁄10ⁿ ; n = 1, 2, 3, …
Problem 10.8.9
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 = 1 to ∞) ((−1)ᵏ⁺¹) / (√2ᵏ + ln k)
Problem 10.7.43
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 ∞) 2ᵏ k! / kᵏ
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 10.1.2
Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.
Problem 10.3.41
21–42. Geometric series Evaluate each geometric series or state that it diverges.
41. ∑ (k = 1 to ∞) 4 / 12ᵏ
Problem 10.2.9
6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.
{(−0.7)ⁿ}
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 10.8.47
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (4k)! / (k!)⁴
Problem 10.17
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (k³ᐟ² + 1)
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 10.8.73
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 0 to ∞) k² · 1.001⁻ᵏ
Problem 10.3.13
9–15. Geometric sums Evaluate each geometric sum.
{Use of Tech} ∑ k = 0 to 9 (−3/4)ᵏ
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 10.5.59
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 10.8.67
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 1 to ∞) cot(–1 / j) / 2ʲ
Problem 10.5.53
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 10.5.39
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 10.1.63
61–66. Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.
4 + 0.9 + 0.09 + 0.009 + ⋯
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 10.2.19
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{1 + cos(1⁄n)}
Problem 10.1.7
The first ten terms of the sequence {(1 + 1/10ⁿ)^10ⁿ}∞ ₙ₌₁ are rounded to 8 digits right of the decimal point (see table). Make a conjecture about the limit of the sequence.
n an
1 2.59374246
2 2.70481383
3 2.71692393
4 2.71814593
5 2.71826824
6 2.71828047
7 2.71828169
8 2.71828179
9 2.71828204
10 2.71828203
Problem 10.1.47
45–48. {Use of Tech} Explicit formulas for sequences Consider the formulas for the following sequences {aₙ}ₙ₌₁∞
Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
aₙ = ⁿ² + n ; n = 1, 2, 3, …
Problem 10.2.79
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
{n¹⁰⁰⁰ / 2ⁿ}
Problem 10.7.39
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 ∞) (−1)ᵏ k³ / √(k⁸ + 1)
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)