14. Sequences & Series

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.

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

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. 

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

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

∑ (from k = 1 to ∞)5¹⁻²ᵏ

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))

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))

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k¹/ᵏ

Use the Cauchy condensation test from Exercise 59 to show that:

b. ∑ (from n=1 to ∞) [1 / nᵖ] converges if p > 1 and diverges if p ≤ 1.

Use the Alternating Series Test to determine the convergence or divergence of the series.

${\sum }_{n=1}^{\infty }\frac{{(-1)}^{n+1}}{\sqrt{n}+2}$

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 ∞) k¹⁰⁰ / (k + 1)!