14. Sequences & Series

Explain why the magnitude of the remainder in an alternating series (with terms that are nonincreasing in magnitude) is less than or equal to the magnitude of the first neglected term.

b. From Example 5, Section 10.2, show that

S = 1 + ∑(from n=1 to ∞) [1 / (n²(n + 1))].

39–44. {Use of Tech} Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 10⁻³.

∑ (k = 1 to ∞) (−1)ᵏ / kᵏ

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.

∑ (from k = 1 to ∞)2ᵏ / 3ᵏ⁺²

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

e. If ∑ k⁻ᵖ converges, then ∑ k⁻ᵖ⁺⁰.⁰⁰¹ converges.

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.

π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

Find the sum of each series in Exercises 45–52.

∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]

71. Evaluating an infinite series two ways

Evaluate the series

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

a. Use a telescoping series argument.

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)]

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 0 to ∞) k / (2k + 1)

{Use of Tech} For what value of r does

∑ (k = 3 to ∞) r²ᵏ = 10?

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.

a. Use Sₙ to estimate the sum of the series.

39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

89–90. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, complete the following.

b. Find an upper bound for the remainder Rₙ.

89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).

51.0.456̅ = 0.456456456…

21–42. Geometric series Evaluate each geometric series or state that it diverges.  

27.1 + 1.01 + 1.01² + 1.01³ + ⋯