14. Sequences & Series

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

∑ (k = 1 to ∞) (−1)ᵏ · k / (2k + 1)

Determine whether the given series are convergent using the Ratio Test.

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

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

∑ (from k = 1 to ∞)(1 − cos(1 / k))²

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

∑ (from k = 1 to ∞)tanh(k)

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

∑ (k = 1 to ∞) (k² − 1) / (k³ + 4)

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

${\sum }_{n=1}^{\infty }\frac{n^2}{n(n^2-1000)}$

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

∑ (from k = 1 to ∞)k⁴ / √(9k¹² + 2)

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

e. ∑ (k = 1 to ∞) (π / e)⁻ᵏ is a convergent geometric series.

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

∑ (from k = 1 to ∞)(cos(1 / k) – cos(1 / (k + 1)))

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

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

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

∑ (from k = 1 to ∞) 2⁹k / kᵏ

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 ∞) ((-7)ᵏ / k²)

{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is

J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ

b. Find the radius and interval of convergence of the power series for J₀.

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))]

What test is advisable if a series involves a factorial term?