14. Sequences & Series

Express 0.314141414… as a ratio of two integers.

Give an example (if possible) of a sequence {aₖ} that converges, while the series ∑ (from k = 1 to ∞) aₖ diverges.

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.

67. ∑ (k = 1 to ∞) 3 / (k² + 5k + 4)

87. Explain why or why not

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

d. If ∑ pᵏ diverges, then ∑ (p + 0.001)ᵏ diverges, for a fixed real number p.

87. Explain why or why not

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

c. If ∑ aₖ converges, then ∑ (aₖ + 0.0001) converges.

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

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ · tan⁻¹(k) / k³

8–32. {Use of Tech} Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − Sₙ| in using the nth partial sum Sₙ to estimate the value of the series S.

∑ (k = 1 to ∞) (−1)ᵏ / k⁴; n = 4

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

c. Suppose f is a continuous, positive, decreasing function, for x ≥ 1, and aₖ = f(k), for k = 1, 2, 3, …. If ∑ (k = 1 to ∞) aₖ converges to L, then ∫ (1 to ∞) f(x) dx converges to L.

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.

63. ∑ (k = 1 to ∞) 1 / ((k + p)(k + p + 1)), where p is a positive integer

Explain why or why not

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

a.The sequence of partial sums for the series1 + 2 + 3 + ⋯ is {1, 3, 6, 10, …}.

Harmonic sum In Chapter 10, we will encounter the harmonic sum 1 + 1/2 + 1/3 + ⋯ + 1/n. Use a left Riemann sum to approximate ∫[1 to n+1] (dx/x) (with unit spacing between the grid points) to show that 1 + 1/2 + 1/3 + ⋯ + 1/n > ln(n + 1). Use this fact to conclude that lim (n → ∞) (1 + 1/2 + 1/3 + ⋯ + 1/n) does not exist.

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

29.∑ (k = 1 to ∞) e^(–2k)

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 ∞) 1 / (1000 + k)

16–17. {Use of Tech} Periodic savings

Suppose you deposit m dollars at the beginning of every month in a savings account that earns a monthly interest rate of r, which is the annual interest rate divided by 12 (for example, if the annual interest rate is 2.4%, r = 0.024/12 = 0.002). For an initial investment of m dollars, the amount of money in your account at the beginning of the second month is the sum of your second deposit and your initial deposit plus interest, or m + m(1 + r). Continuing in this fashion, it can be shown that the amount of money in your account after n months is

Aₙ = m + m(1 + r) + ⋯ + m(1 + r)ⁿ⁻¹.

Use geometric sums to determine the amount of money in your savings account after 5 years (60 months) using the given monthly deposit and interest rate.

17. Monthly deposits of \$250 at a monthly interest rate of 0.2%