72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence


b.Find an explicit formula for the terms of the sequence.


Radioactive decay
A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

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

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

b. Which of the following series converges faster? Explain.

∑ (k = 2 to ∞) 1 / (k(ln k)²) or ∑ (k = 3 to ∞) 1 / (k(ln k)(ln ln k)²)?

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

b. A series that converges absolutely must converge.

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

{-5, 5, -5, 5, ......}

71. Evaluating an infinite series two ways

Evaluate the series

∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.

b. Use a geometric series argument with Theorem 10.8.

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.