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.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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

∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)

Define the remainder of an infinite series.

Give an example (if possible) of a sequence {aₖ} that converges, while the series ∑ (from k = 1 to ∞) aₖ diverges.

Finding steady states using infinite series Solve Exercise 40 by expressing the amount of aspirin in your blood as a geometric series and evaluating the series.