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문제 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
문제 10.2.43
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{√((1 + 1 / 2n)ⁿ)}
문제 10.1.23
"21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 3aₙ-12; a₁ = 10
문제 10.4.19
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))
문제 10.1.17
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)
문제 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
문제 10.6.35
33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.
π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)
문제 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)))
문제 10.8.29
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰
문제 10.8.49
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (⁵√k) / ⁵√(k⁷ + 1)
문제 10.1.21
21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 2aₙ; a₁ = 2
문제 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
문제 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)
문제 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!)
문제 10.7.47
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
1 / 1! + 4 / 2! + 9 / 3! + 16 / 4! + ⋯
문제 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)
문제 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)}
문제 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))
문제 10.3.91
{Use of Tech} For what value of r does
∑ (k = 3 to ∞) r²ᵏ = 10?
문제 10.3.75
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)
문제 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
문제 10.8.7
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 ∞) (k² / (k⁴ + k³ + 1))
문제 10.6.39
39–44. {Use of Tech} Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 10⁻³.
∑ (k = 1 to ∞) (−1)ᵏ / k⁵
문제 10.3.35
21–42. Geometric series Evaluate each geometric series or state that it diverges.
35. ∑ (k = 0 to ∞) 3(–π)^(–k)
문제 10.4.13
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 / ln k
문제 10.2.31
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(3ⁿ⁺¹ + 3)⁄3ⁿ}
문제 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)
문제 10.8.35
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2⁹k / kᵏ
문제 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
문제 10.R.33
27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 0 to ∞) (tan⁻¹(k + 2) − tan⁻¹k)