Suppose the sequence {aₙ}⁽∞⁾ₙ₌₀ is defined by the recurrence relation
aₙ₊₁ = ⅓aₙ + 6;a₀ = 3.


b.Explain why {aₙ}⁽∞⁾ₙ₌₀ converges and find the limit.

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

b.Find a formula for the nth partial sum Sₙ of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, S₈ of the infinite series.

∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

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

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

b. (2n)! / (2n − 1)! = 2n

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).

{1, 3, 9, 27, 81, ......}

87. Explain why or why not

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

b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.

39–40. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, use Theorem 10.13 to complete the following.

b. Find an upper bound for the remainder Rₙ.

39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2