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8:22{Use of Tech} A savings plan
James begins a savings plan in which he deposits \(100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.
To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \)100.
Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \$0.
b.Find a recurrence relation that generates the sequence {Bₙ}.
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
b.If limₙ→∞aₙ = 0 and limₙ→∞bₙ = ∞, then limₙ→∞aₙbₙ = 0.
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
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.
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)?
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
