14. Sequences & Series

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

a. Find an upper bound for the remainder in terms of n.

41. ∑ (k = 1 to ∞) 1 / k⁶

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

33.∑ (k = 4 to ∞) 1 / 5ᵏ

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

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

39.∑ (k = 2 to ∞) (–0.15)ᵏ

Does a geometric series always have a finite value?

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

25.∑ (k = 0 to ∞) 0.9ᵏ

Error Estimation

In Exercises 49–52, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.

1 / (1 + t) = ∑ (from n = 0 to ∞) [(-1)ⁿ tⁿ],0 < t < 1

Compute the first four partial sums and find a formula for the $n^{\mathrm{th}}$ partial sum.

${\sum }_{n=1}^{\infty }2n-1$

Estimate the value of the series ∑ (from k = 1 to ∞)1 / (2k + 5)³ to within 10⁻⁴ of its exact value.

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

f. If lim (k → ∞) aₖ = 0, then ∑ aₖ converges."

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 2 to ∞) k / ln k

9–15. Geometric sums Evaluate each geometric sum.

{Use of Tech}∑ k = 0 to 20(2/5)²ᵏ

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

∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)

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

35.∑ (k = 0 to ∞) 3(–π)^(–k)

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

∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)