14. Sequences & Series

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

33.∑ (k = 4 to ∞) 1 / 5ᵏ

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

35.∑ (k = 0 to ∞) 3(–π)^(–k)

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

37.1 + e/π + e²/π² + e³/π³ + ⋯

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

39.∑ (k = 2 to ∞) (–0.15)ᵏ

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

41.∑ (k = 1 to ∞) 4 / 12ᵏ

61–66. Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.

4 + 0.9 + 0.09 + 0.009 + ⋯  

Geometric sums

Evaluate the geometric sums

∑ (from k = 0 to 9) (0.2)ᵏand∑ (from k = 2 to 9) (0.2)ᵏ.

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.

Is it possible for a series of positive terms to converge conditionally? Explain.

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…

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…

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))

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))

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)

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