14. Sequences & Series

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…

71. Evaluating an infinite series two ways

Evaluate the series

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

a. Use a telescoping series argument.

46–53. Decimal expansions

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

53.0.00952̅ = 0.00952952…

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

∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)

Find the first term a and the ratio r of each geometric series.

a. ∑ k = 0 to ∞(2/3) × (1/5)ᵏ

9–15. Geometric sums Evaluate each geometric sum.

∑ k = 0 to 83ᵏ

9–15. Geometric sums Evaluate each geometric sum.

{Use of Tech}∑ k = 0 to 20(2/5)²ᵏ

9–15. Geometric sums Evaluate each geometric sum.

{Use of Tech}∑ k = 0 to 9(−3/4)ᵏ

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

21.∑ (k = 0 to ∞) (1/4)ᵏ

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

23.∑ (k = 0 to ∞) (–9/10)ᵏ

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

25.∑ (k = 0 to ∞) 0.9ᵏ

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

27.1 + 1.01 + 1.01² + 1.01³ + ⋯

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

29.∑ (k = 1 to ∞) e^(–2k)

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

31.∑ (k = 1 to ∞) 2^(–3k)

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

33.∑ (k = 4 to ∞) 1 / 5ᵏ