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Problem 10.3.9
9–15. Geometric sums Evaluate each geometric sum.
∑ k = 0 to 8 3ᵏ
Problem 10.3.11
9–15. Geometric sums Evaluate each geometric sum.
{Use of Tech} ∑ k = 0 to 20 (2/5)²ᵏ
Problem 10.3.13
9–15. Geometric sums Evaluate each geometric sum.
{Use of Tech} ∑ k = 0 to 9 (−3/4)ᵏ
Problem 10.3.21
21–42. Geometric series Evaluate each geometric series or state that it diverges.
21. ∑ (k = 0 to ∞) (1/4)ᵏ
Problem 10.3.23
21–42. Geometric series Evaluate each geometric series or state that it diverges.
23. ∑ (k = 0 to ∞) (–9/10)ᵏ
Problem 10.3.25
21–42. Geometric series Evaluate each geometric series or state that it diverges.
25. ∑ (k = 0 to ∞) 0.9ᵏ
Problem 10.3.27
21–42. Geometric series Evaluate each geometric series or state that it diverges.
27. 1 + 1.01 + 1.01² + 1.01³ + ⋯
Problem 10.3.29
21–42. Geometric series Evaluate each geometric series or state that it diverges.
29. ∑ (k = 1 to ∞) e^(–2k)
Problem 10.3.31
21–42. Geometric series Evaluate each geometric series or state that it diverges.
31. ∑ (k = 1 to ∞) 2^(–3k)
Problem 10.3.33
21–42. Geometric series Evaluate each geometric series or state that it diverges.
33. ∑ (k = 4 to ∞) 1 / 5ᵏ
Problem 10.3.35
21–42. Geometric series Evaluate each geometric series or state that it diverges.
35. ∑ (k = 0 to ∞) 3(–π)^(–k)
Problem 10.3.37
21–42. Geometric series Evaluate each geometric series or state that it diverges.
37. 1 + e/π + e²/π² + e³/π³ + ⋯
Problem 10.3.39
21–42. Geometric series Evaluate each geometric series or state that it diverges.
39. ∑ (k = 2 to ∞) (–0.15)ᵏ
Problem 10.3.41
21–42. Geometric series Evaluate each geometric series or state that it diverges.
41. ∑ (k = 1 to ∞) 4 / 12ᵏ
Problem 10.3.7
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.
Problem 10.3.49
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…
Problem 10.3.51
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…
Problem 10.3.57
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))
Problem 10.3.59
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))
Problem 10.3.61
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)
Problem 10.3.63
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
Problem 10.3.65
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.
65. ∑ (k = 1 to ∞) (1 / √(k + 1) – 1 / √(k + 3))
Problem 10.3.67
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)
Problem 10.3.3
Does a geometric series always have a finite value?
Problem 10.3.87a
87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ converges.
Problem 10.3.87b
87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.
Problem 10.3.87c
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.
Problem 10.3.87e
87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
e. ∑ (k = 1 to ∞) (π / e)⁻ᵏ is a convergent geometric series.
Problem 10.3.87g
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, ∞).
Problem 10.3.87d
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.