Back
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 10.2.43
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{√((1 + 1 / 2n)ⁿ)}
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 10.4.19
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 10.1.17
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (2ⁿ⁺¹) / (2ⁿ + 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 10.6.35
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 10.8.65
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 10.8.29
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰
Problem 10.8.49
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (⁵√k) / ⁵√(k⁷ + 1)
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 10.3.75
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)
Problem 10.4.51
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 ∞) 1 / ( (3k + 1)(3k + 4) )
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 10.7.47
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! + ⋯
Problem 10.2.81
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)
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 10.7.33
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 10.3.91
{Use of Tech} For what value of r does
∑ (k = 3 to ∞) r²ᵏ = 10?
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 10.3.29
21–42. Geometric series Evaluate each geometric series or state that it diverges.
29. ∑ (k = 1 to ∞) e^(–2k)
Problem 10.4.11
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 ∞) 1 / (1000 + k)
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 10.4.49
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 10.8.71
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (ln²k) / k³ᐟ²
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 10.2.31
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(3ⁿ⁺¹ + 3)⁄3ⁿ}
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 10.6.51
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)