14. Sequences & Series

Is it possible for an alternating series to converge absolutely but not conditionally?

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

c. The terms of the sequence of partial sums of the series ∑ aₖ approach 5/2, so the infinite series converges to 5/2.

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.

∑ (k = 1 to ∞) 1 / ( (3k + 1)(3k + 4) )

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

d. Every partial sum Sₙ of the series ∑ (k = 1 to ∞) 1 / k² underestimates the exact value of ∑ (k = 1 to ∞) 1 / k².

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

c. Suppose f is a continuous, positive, decreasing function, for x ≥ 1, and aₖ = f(k), for k = 1, 2, 3, …. If ∑ (k = 1 to ∞) aₖ converges to L, then ∫ (1 to ∞) f(x) dx converges to L.

67–70. Formulas for sequences of partial sums Consider the following infinite series.

a.Find the first four partial sums S₁, S₂, S₃, S₄ of the series.

∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

67–70. Formulas for sequences of partial sums Consider the following infinite series.

c.Make a conjecture for the value of the series.

∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

Explain why or why not

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

a.The sequence of partial sums for the series1 + 2 + 3 + ⋯ is {1, 3, 6, 10, …}.

Explain why or why not

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

c.If the terms of the sequence {aₙ} are positive and increasing, then the sequence of partial sums for the series∑⁽∞⁾ₖ₌₁aₖ diverges.

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

a. The sum ∑ (k = 1 to ∞) 1 / 3ᵏ is a p-series.

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

b. The sum ∑ (k = 3 to ∞) 1 / √(k − 2) is a p-series.

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

∑ (k = 1 to ∞) (−1)ᵏ · tan⁻¹(k) / k³

Property of divergent series Prove Property 2 of Theorem 10.8: If ∑ aₖ diverges, then ∑ caₖ also diverges, for any real number c ≠ 0.

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

a. Find an upper bound for the remainder in terms of n.

41. ∑ (k = 1 to ∞) 1 / k⁶

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.

41. ∑ (k = 1 to ∞) 1 / k⁶