14. Sequences & Series

Repeating Decimals

Express each of the numbers in Exercises 23–30 as the ratio of two integers.

3.1̅4̅2̅8̅5̅7 = 3.142857142857 ...

46–53. Decimal expansions

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

49.0.037̅ = 0.037037…

∑ (from n=1 to ∞) (1 / √(n + 1)) diverges

b. What should n be in order that the partial sum sₙ = ∑ (from i=1 to n) (1 / √(i + 1)) satisfies sₙ > 1000?

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.

57. ∑ (k = 1 to ∞) 1 / ((k + 6)(k + 7))

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

41.∑ (k = 1 to ∞) 4 / 12ᵏ

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

a. Find an upper bound for the remainder in terms of n.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.

b. The expected number of rounds (possessions by either team) required for the overtime to end is (1/6) ∑ₖ₌₁∞ k(5/6)ᵏ⁻¹. Evaluate this series.

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.

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.

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

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.

ln 2 = ∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / k

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

Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.

a. The probability that Team A ultimately wins is ∑ₖ₌₀∞ (1/6)(5/6)²ᵏ. Evaluate this series.

87. Explain why or why not

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

g. Viewed as a function of r, the series 1 + r + r² + r³ + ⋯ takes on all values in the interval (1/2, ∞).

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.

41. ∑ (k = 1 to ∞) 1 / k⁶

Property of divergent series Prove Property 2 of Theorem 10.8: If ∑ aₖ diverges, then ∑ caₖ also diverges, for any real number c ≠ 0.