14. Sequences & Series

Does the series

∑ (from n=1 to ∞) (1/n − 1/n²)

converge or diverge? Justify your answer.

Use the Limit Comparison Test to determine if the following series converges.

${\sum }_{n=1}^{\infty }\frac{1}{2n^2-n\cos \left(n\pi \right)}$

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

∑ (from k = 1 to ∞) (2k⁴ + k) / (4k⁴ − 8k)

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

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

∑ (from k = 1 to ∞)(⁵√k) / ⁵√(k⁷ + 1)

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))

Limit Comparison Test

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

∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))

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 + ⋯

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

∑ (from k = 3 to ∞) 5 / (2 + lnk)

If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.

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

∑ (from k = 1 to ∞) 5ᵏ(k!)² / (2k)!

Direct Comparison Test

In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.

∑ (from n=1 to ∞) cos²n / n^(3/2)

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) 4ᵏ / (5ᵏ − 3)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A series that converges must converge absolutely.

Determine the convergence or divergence of the series. 

${\sum }_{n=1}^{\infty }\frac{n\bullet 8^n}{(n+1)!}$