13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(3n³ − 1)⁄(2n³ + 1)}

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 3 to ∞) 5 / (2 + ln k)

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 1 / (√k × e^(√k))

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.

2 / 4² + 2 / 5² + 2 / 6² + ⋯

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 ∞) (√k / k − 1)²ᵏ

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, …

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 2ᵏ / (3ᵏ − 2ᵏ)

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.

Is it possible for an alternating series to converge absolutely but not conditionally?

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 ∞) 3ᵏ / (k² + 1)

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 0 to ∞) 3k / ∜(k⁴ + 3)

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))

Series of squares Prove that if ∑ aₖ is a convergent series of positive terms, then the series ∑ aₖ² also converges.

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{tan⁻¹(n)}

84–87. {Use of Tech} Sequences by recurrence relations

The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.

a. Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.

b. Use analytical methods to find the limit of the sequence.

{Use of Tech} aₙ₊₁ = √(2 + aₙ); a₀ = 3

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 5ᵏ(k!)² / (2k)!

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 ∞) (5 / 6)⁻ᵏ

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(3ⁿ⁺¹ + 3)⁄3ⁿ}

84–87. {Use of Tech} Sequences by recurrence relations

The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.

a. Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.

b. Use analytical methods to find the limit of the sequence.

aₙ₊₁ = 2aₙ (1 − aₙ); a₀ = 0.3

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(1 + (2 / n))ⁿ}

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. 

aₙ = 1⁄10ⁿ ; n = 1, 2, 3, …

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

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))

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 5¹⁻²ᵏ

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 ∞) ((k / (k + 1)) × 2k²)

Compare the growth rates of {n¹⁰⁰} and {eⁿ⁄¹⁰⁰} as n → ∞.

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.

65. ∑ (k = 1 to ∞) (1 / √(k + 1) – 1 / √(k + 3))

46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).

53. 0.00952̅ = 0.00952952…

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 2⁹k / kᵏ

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) cos(k) / k³