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.  

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

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.  

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.