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Problem 10.17
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (k³ᐟ² + 1)
Problem 10.2.55
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(−1)ⁿ / 2ⁿ}
Problem 10.2.73a
{Use of Tech} A savings plan
James begins a savings plan in which he deposits $100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.
To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits $100.
Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = $0.
a. Write the first five terms of the sequence {Bₙ}.
Problem 10.2.13
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{n³⁄(n⁴ + 1)}
Problem 10.2.15
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(3n³ − 1)⁄(2n³ + 1)}
Problem 10.2.17
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹(10n⁄(10n + 4))}
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 10.2.23
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(√(4n⁴ + 3n))⁄(8n² + 1)}
Problem 10.2.27
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{2ⁿ⁺¹3⁻ⁿ}
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 10.2.33
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(n + 1)!⁄n!}
Problem 10.2.35
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹(n)}
Problem 10.2.51
Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{n sin(6 / n)}
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 10.2.59
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{n sin³(nπ / 2) / (n + 1)}"
Problem 10.2.61
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
aₙ = e⁻ⁿ cos n
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 10.2.65
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{cos n / n}
Problem 10.2.67
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{sin n / 2ⁿ}
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 10.2.37
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{√(n² + 1) − n}
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 10.2.41
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{ⁿ√(e³ⁿ⁺⁴)}
Problem 10.2.77
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
{n¹⁰ / ln 20 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 10.2.81
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 10.2.83a
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
a. If limₙ→∞ aₙ = 1 and limₙ→∞ bₙ = 3, then limₙ→∞ (bₙ / aₙ) = 3.
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 10.2.47
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(1 / n)¹⁄ⁿ}
Problem 10.2.95a
{Use of Tech} Repeated square roots
Consider the sequence defined by
aₙ₊₁ = √(2 + aₙ), a₀ = √2, for n = 0, 1, 2, 3, …
a. Evaluate the first four terms of the sequence {aₙ}.
State the exact values first, and then the approximate values.