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Problem 10.3.21
21–42. Geometric series Evaluate each geometric series or state that it diverges.
21. ∑ (k = 0 to ∞) (1/4)ᵏ
Problem 10.7.13
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 ∞) (k² / 4ᵏ)
Problem 10.2.5
For what values of r does the sequence {rⁿ} converge? Diverge?
Problem 10.8.45
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) k⁴ / (eᵏ⁵)
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 10.8.11
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (2k⁴ + k) / (4k⁴ − 8k)
Problem 10.3.79
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 2 to ∞) ln((k + 1)k⁻¹) / (ln k × ln(k + 1))
Problem 10.6.29
8–32. {Use of Tech} Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − Sₙ| in using the nth partial sum Sₙ to estimate the value of the series S.
∑ (k = 1 to ∞) (−1)ᵏ / k⁴; n = 4
Problem 10.3.97
Property of divergent series Prove Property 2 of Theorem 10.8: If ∑ aₖ diverges, then ∑ c aₖ also diverges, for any real number c ≠ 0.
Problem 10.8.61
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 1 / ln(eᵏ + 1)
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 10.1.17
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)
Problem 10.4.15
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 / (ln¹⁰ k)
Problem 10.3.17
16–17. {Use of Tech} Periodic savings
Suppose you deposit m dollars at the beginning of every month in a savings account that earns a monthly interest rate of r, which is the annual interest rate divided by 12 (for example, if the annual interest rate is 2.4%, r = 0.024/12 = 0.002). For an initial investment of m dollars, the amount of money in your account at the beginning of the second month is the sum of your second deposit and your initial deposit plus interest, or m + m(1 + r). Continuing in this fashion, it can be shown that the amount of money in your account after n months is
Aₙ = m + m(1 + r) + ⋯ + m(1 + r)ⁿ⁻¹.
Use geometric sums to determine the amount of money in your savings account after 5 years (60 months) using the given monthly deposit and interest rate.
17. Monthly deposits of $250 at a monthly interest rate of 0.2%
Problem 10.6.13
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)
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 10.2.79
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
{n¹⁰⁰⁰ / 2ⁿ}
Problem 10.6.7
Is it possible for a series of positive terms to converge conditionally? Explain.
Problem 10.7.45
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 / k⁰.⁹⁹
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 10.5.50
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 2 to ∞) (5 ln k) / k
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 10.2.81
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 10.3.11
9–15. Geometric sums Evaluate each geometric sum.
{Use of Tech} ∑ k = 0 to 20 (2/5)²ᵏ
Problem 10.3.29
21–42. Geometric series Evaluate each geometric series or state that it diverges.
29. ∑ (k = 1 to ∞) e^(–2k)
Problem 10.5.21
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) ((3k³ + 4)(7k² + 1)) / ((2k³ + 1)(4k³ − 1))
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 10.2.43
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{√((1 + 1 / 2n)ⁿ)}
Problem 10.2.33
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