14. Sequences & Series

Limit Comparison Test

In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.

∑ (from n=2 to ∞) 1 / ln n

(Hint: Limit Comparison with ∑ (from n=2 to ∞) (1/n))

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) 1 / (2√n + ³√n)

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (n / (3n + 1))ⁿ

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (1 − n) / n2ⁿ

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) sin (1/n)

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (tanh n) / n²

Using the Ratio Test

In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.

∑(from n=1 to ∞) [(-1)ⁿ (n + 2) / 3ⁿ]

Using the Ratio Test

In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.

∑(from n=2 to ∞) [(3ⁿ⁺²) / ln(n)]

Using the Root Test

In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.

∑(from n=1 to ∞) [4ⁿ / (3n)ⁿ]

Using the Root Test

In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.

∑(from n=1 to ∞) [(-1)ⁿ (1 − 1/n)ⁿ^²]

(Hint: lim (n→∞) (1 + x/n)ⁿ = eˣ)

Determining Convergence or Divergence

In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑(from n=1 to ∞) [(-1)ⁿ n² e⁻ⁿ]

Convergence or Divergence

Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]

Assume that bₙ is a sequence of positive numbers converging to 4/5. Determine if the following series converge or diverge.

b. ∑ (from n = 1 to ∞) (5/4)ⁿ (bₙ)

Determining Convergence or Divergence

In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]