BackExplain 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
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.
41. ∑ (k = 1 to ∞) 1 / k⁶
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.
41. ∑ (k = 1 to ∞) 1 / k⁶
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.
43. ∑ (k = 1 to ∞) 1 / 3ᵏ