14. Sequences & Series

b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?

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

∑ (k = 1 to ∞) (2 + (−1)ᵏ) / k²

Explain why the integral test does not apply to the series.

${\sum }_{n=4}^{\infty }\frac{1}{n+\sin n}$

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = 8^(1/n)

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (2n + 2)! / (2n − 1)!

Explain why the integral test does not apply to the series.

${\sum }_{n=1}^{\infty }\frac{{(-1)}^n}{\ln n}$

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

{Use of Tech} ∑ (k = 1 to ∞) 1 / (4lnk)

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

d. If ∑ aₖ diverges, then ∑ |aₖ| diverges.

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

∑ (from k = 1 to ∞)1 / ln(eᵏ + 1)

Loglog p-series Consider the series ∑ (k = 2 to ∞) 1 / (k(ln k)(ln ln k)ᵖ), where p is a real number.

a. For what values of p does this series converge?

Absolute and Conditional Convergence

Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ⁻¹ / (n² + 2n + 1)]

Determine the convergence or divergence of the series. 

${\sum }_{k=1}^{\infty }\frac{{(-1)}^{k}4}{3k+2}$

Use the Alternating Series Test to determine if the following series is conditionally convergent, absolutely convergent, or divergent.

${\sum }_{k=1}^{\infty }\frac{\sin \frac{n\pi }{2}}{n^3}$

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 ∞) (2ᵏ / k⁹⁹)

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

∑ (from k = 1 to ∞)2ᵏ / eᵏ