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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 10.1.3
Suppose the sequence { aₙ} is defined by the recurrence relation a₍ₙ₊₁₎ = n · aₙ , for n=1, 2, 3 ...., where a₁ = 1. Write out the first five terms of the sequence.
Problem 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
Problem 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
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 10.8.65
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 10.6.35
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)
Problem 10.3.35
21–42. Geometric series Evaluate each geometric series or state that it diverges.
35. ∑ (k = 0 to ∞) 3(–π)^(–k)
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 10.2.63
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹ n / n}
Problem 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
Problem 10.2.37
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{√(n² + 1) − n}
Problem 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)
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 10.8.51
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (7ᵏ + 11ᵏ) / 11ᵏ
Problem 10.2.39
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(1 + (2 / n))ⁿ}
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 10.8.9
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 10.7.37
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 ∞) k¹⁰⁰ / (k + 1)!
Problem 10.6.61
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)
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 10.8.7
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))
Problem 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))
Problem 10.1.43
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.
{Use of Tech} aₙ₊₁ = (aₙ⁄₁₁ )+ 50 ; a₀ = 50
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 10.2.49
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{bₙ}, where
bₙ = { n / (n + 1) if n ≤ 5000
ne⁻ⁿ if n > 5000 }
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 10.2.33
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(n + 1)!⁄n!}
Problem 10.6.51
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) cos(k) / k³
Problem 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)