14. Sequences & Series

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

c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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

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

c. Find lower and upper bounds (Lₙ and Uₙ respectively) for the exact value of the series.

89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

Find a formula for the nth partial sum Sₙ of

∑ k = 1 to ∞[(1/(k + 3)) − (1/(k + 4))]

Use your formula to find the sum of the first 36 terms of the series.

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) (((1/6)ᵏ + (1/3)ᵏ) × k⁻¹)

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

d. Every partial sum Sₙ of the series ∑ (k = 1 to ∞) 1 / k² underestimates the exact value of ∑ (k = 1 to ∞) 1 / k².

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

37.1 + e/π + e²/π² + e³/π³ + ⋯

46–53. Decimal expansions

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

47.0.3̅ = 0.333…

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

c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.

41. ∑ (k = 1 to ∞) 1 / k⁶

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.

61. ∑ (k = 1 to ∞) ln((k + 1) / k)

Find the value of b for which

1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.

Why does the value of a converging alternating series with terms that are nonincreasing in magnitude lie between any two consecutive terms of its sequence of partial sums?

Building a tunnel — first scenario

A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.

b.What is the longest tunnel the crew can build at this rate?

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 / k⁵