28. IM. IM. M. sin n 2" 30. (-1)-1. 32. (-1)". (-1)" n-2 n ln n 34. Σ n n² + 4 n √n³+2 Tod 29. 31. 33. n=1 1 + 2 sin n (-1)" n-1 n+p COS NTT n-1 3n+2 47. Σ 48. Σ(-1)-1 n-2 (In n) (-1)" sb ni tuleau vipy 49. Show that the series Σ(-1)"bn, where b = 1/n if n is odd and bn = 1/n² if n is even, is divergent. Why does the Alter- nating Series Test not apply? n-2 ln n 50. Use the following steps to show that (-1)"-¹ Σ n=1 n = In 2 n e sole Let h.. and s.. be the partial sums of the harmonic and alter-
28. IM. IM. M. sin n 2" 30. (-1)-1. 32. (-1)". (-1)" n-2 n ln n 34. Σ n n² + 4 n √n³+2 Tod 29. 31. 33. n=1 1 + 2 sin n (-1)" n-1 n+p COS NTT n-1 3n+2 47. Σ 48. Σ(-1)-1 n-2 (In n) (-1)" sb ni tuleau vipy 49. Show that the series Σ(-1)"bn, where b = 1/n if n is odd and bn = 1/n² if n is even, is divergent. Why does the Alter- nating Series Test not apply? n-2 ln n 50. Use the following steps to show that (-1)"-¹ Σ n=1 n = In 2 n e sole Let h.. and s.. be the partial sums of the harmonic and alter-
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Section 11.5: Alternating Series and Absolute Convergence**
---
**Key Concepts:**
21. **Series Convergence Questions:**
(a) What does it mean for a series to be absolutely convergent?
(b) What does it mean for a series to be conditionally convergent?
(c) If the series of positive terms ∑an converges, what can you say about the series ∑an, where an includes negative terms?
**Exercises:**
22–34. **Determine the nature of convergence for each series:**
22. ∑ (-1)^(n-1) / n^4
23. ∑ (-1)^(n-1) / n^n
24. ∑ (-1)^(n+1) / n + 1
25. ∑ (-1)^(n-1) / (2n - 1)
26. ∑ (-1)^n * n / (n + 1)
27. ∑ (sin n) / n
28. ∑ 1 / (2 + sin n)
29. ∑ (-1)^n / (n + p)
30. ∑ (-1)^n * n / (n^2 + 4)
31. ∑ (-1)^n / ln n
32. ∑ (-1)^n / (n^2 + 3)
33. ∑ (cos nπ) / n + 2
34. ∑ (-1)^(2n) / |2n - 1|
35–36. **Graph Analysis:**
Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to estimate the sum of the series with the Alternating Series Estimation Theorem to four decimal places.
35. ∑ (-0.8)^n / n!
36. ∑ (-1)^(n-1) * n / 8^n
37–40. **Convergence Accuracy:**
Show that the series is convergent. Determine how many terms are required to find the sum with the specified accuracy.
37. ∑ (-1)^(n+1) / n^6 [error < 0.00005]
38. ∑ (-1)^(n+1) / n^2 [error < 0.0005]
39. ∑ (-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7534f13a-064e-4169-8c87-38e5472c444f%2F6ae958fa-3975-4809-9a5b-dfd65952ebb9%2Frxry5zf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Section 11.5: Alternating Series and Absolute Convergence**
---
**Key Concepts:**
21. **Series Convergence Questions:**
(a) What does it mean for a series to be absolutely convergent?
(b) What does it mean for a series to be conditionally convergent?
(c) If the series of positive terms ∑an converges, what can you say about the series ∑an, where an includes negative terms?
**Exercises:**
22–34. **Determine the nature of convergence for each series:**
22. ∑ (-1)^(n-1) / n^4
23. ∑ (-1)^(n-1) / n^n
24. ∑ (-1)^(n+1) / n + 1
25. ∑ (-1)^(n-1) / (2n - 1)
26. ∑ (-1)^n * n / (n + 1)
27. ∑ (sin n) / n
28. ∑ 1 / (2 + sin n)
29. ∑ (-1)^n / (n + p)
30. ∑ (-1)^n * n / (n^2 + 4)
31. ∑ (-1)^n / ln n
32. ∑ (-1)^n / (n^2 + 3)
33. ∑ (cos nπ) / n + 2
34. ∑ (-1)^(2n) / |2n - 1|
35–36. **Graph Analysis:**
Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to estimate the sum of the series with the Alternating Series Estimation Theorem to four decimal places.
35. ∑ (-0.8)^n / n!
36. ∑ (-1)^(n-1) * n / 8^n
37–40. **Convergence Accuracy:**
Show that the series is convergent. Determine how many terms are required to find the sum with the specified accuracy.
37. ∑ (-1)^(n+1) / n^6 [error < 0.00005]
38. ∑ (-1)^(n+1) / n^2 [error < 0.0005]
39. ∑ (-
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