14. Sequences & Series

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)5ᵏ / 2²ᵏ⁺¹

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 ∞) ((-1)ᵏ) / (k!)

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

∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰

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 ∞) (−1)ᵏ k (2ᵏ⁺¹ / (9ᵏ − 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 ∞) ((−1)ᵏ⁺¹) / (√2ᵏ + lnk)

(Continuation of Exercise 61.) Use the result in Exercise 61 to determine which of the following series converge and which diverge. Support your answer in each case.

a. ∑ (from n=2 to ∞) [1 / (n ln n)]

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞)(−1)ᵏk·e⁻ᵏ

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

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.

∑ (from n = 1 to ∞) (1 − 1/n)ⁿ

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

c. A series that converges conditionally must converge.

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

c. If lim (as k → ∞) ᵏ√|aₖ| = 1/4, then ∑ 10aₖ converges absolutely.

Determine whether the given series is convergent.

${\sum }_{n=1}^{\infty }\frac{1}{{}^{{}^5}\sqrt{n^3}}$

Use the divergence test to determine if the following series diverge or state that the test is inconclusive. 

${\sum }_{n=1}^{\infty }\frac{{10}^n}{n!}$

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

∑ (k = 1 to ∞) cos(k) / k³

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

∑ (from k = 1 to ∞)(1 + 1 / (2k))ᵏ