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Problem 10.8.81
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (1 / √(k + 2) – 1 / √k)
Problem 10.7.3
Evaluate 1000!/998! without a calculator.
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 10.1.2
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 10.8.47
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (4k)! / (k!)⁴
Problem 10.3.49
46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).
49. 0.037̅ = 0.037037…
Problem 10.6.57
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ · k / (2k + 1)
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 10.8.65
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 10.3.83
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 1 to ∞) ((1/3) × (5/6)ᵏ + (3/5) × (7/9)ᵏ)
Problem 10.3.57
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.
57. ∑ (k = 1 to ∞) 1 / ((k + 6)(k + 7))
Problem 10.3.31
21–42. Geometric series Evaluate each geometric series or state that it diverges.
31. ∑ (k = 1 to ∞) 2^(–3k)
Problem 10.8.29
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰
Problem 10.2.33
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(n + 1)!⁄n!}
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 10.8.35
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2⁹k / kᵏ
Problem 10.2.35
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹(n)}
Problem 10.7.7
What test is advisable if a series involves a factorial term?
Problem 10.7.33
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 k (2ᵏ⁺¹ / (9ᵏ − 1))
Problem 10.6.49
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (3/4)ᵏ
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 10.3.43
43–44. Periodic doses
Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount of medication in your blood just after your nth dose is
Aₙ = m + mf + ⋯ + mfⁿ⁻¹.
For the given values of f and m, calculate A₅, A₁₀, A₃₀, and lim (n → ∞) Aₙ. Interpret the meaning of the limit lim (n → ∞) Aₙ.
43. f = 0.25, m = 200 mg
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 10.8.73
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 0 to ∞) k² · 1.001⁻ᵏ
Problem 10.7.17
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 10.6.17
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)
Problem 10.3.41
21–42. Geometric series Evaluate each geometric series or state that it diverges.
41. ∑ (k = 1 to ∞) 4 / 12ᵏ
Problem 10.7.9
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) / (k!)
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 10.8.77
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 0 to ∞) 3k / ∜(k⁴ + 3)