Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series ∑ n = 1 ∞ (−1) n + 1 (0.05) n n. The magnitude of the error is approximately ◻. (Simplify your answer. Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to three decimal places as needed.)
Estimate the magnitude of the error involved in using the sum of the first three terms to approximate the sum of the entire series ∑ n = 1 ∞ (−1) n + 1 (0.04) n n. The magnitude of the error is approximately ◻. (Simplify your answer. Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to three decimal places as needed.)
Use the alternating series estimation theorem to determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001. ∑ n = 1 ∞ (−1) n + 1 1 (n+ 2 6n 3 ) 3 or more terms should be used to estimate the sum of the entire series with an error of less than 0.001 .
Use the alternating series estimation theorem to determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001. ∑ n = 1 ∞ (−1) n + 1 1 (n + 5n) 3 or more terms should be used to estimate the sum of the entire series with an error of less than 0.001.
Approximate the sum with an error of magnitude less than 5×10 −6 . ∑ n = 0 ∞ (−1) n 1 (3n)! The sum is approximately ◻. (Round to five decimal places as needed.)
Approximate the sum with an error of magnitude less than 5×10 −7. ∑ n = 0 ∞ (−1) n 1 (5n)! The sum is approximately ◻. (Round to five decimal places as needed.)
Determine whether the series ∑ n = 0 ∞ sin(18n + 1)π 2 converges or diverges. If it converges, find its sum. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The series converges because lim n → ∞ sin(18n + 1)π 2 = 0 . The sum of the series is ◻. (Type an exact answer, using radicals as needed.) B. The series diverges because it is a geometric series with |r| ≥ 1. c. The series converges because it is a geometric series with |r| < 1. The sum of the series is ◻. (Type an exact answer, using radicals as needed.) D. The series diverges because lim n → ∞ sin(18n + 1)π 2 ≠ 0 or fails to exist. E. The series converges because lim k → ∞ ∑ n = 0 k sin(18n + 1)π 2 fails to exist.
Does the series ∑ n = 1 ∞ (−1) n + 1 n 3 n 6 + 1 converge absolutely, converge conditionally, or diverge? Choose the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The series converges absolutely because the limit used in the nth-Term Test is ◻. B. The series diverges because the limit used in the Ratio Test is not less than or equal to 1. C. The series converges conditionally per the Alternating Series Test and the Comparison Test with ∑ n = 1 ∞ 1 n 3. D. The series converges conditionally per the Alternating Series Test and because the limit used in the Root Test is ◻. E. The series converges absolutely per the Comparison Test with ∑ n = 1 ∞ 1 n 3. F. The series diverges because the limit used in the nth-Term Test is not zero.
Does the series ∑ n = 1 ∞ (−1) n 1 n+6 converge absolutely, converge conditionally, or diverge? Choose the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The series diverges because the limit used in the nth-Term Test is not zero. B. The series converges absolutely since the corresponding series of absolute values is geometric with |r| = ◻. C. The series converges conditionally per the Alternating Series Test and the Limit Comparison Test with ∑ n = 1 ∞ 1 n. D. The series converges absolutely since the corresponding series of absolute values is the p-series with p = ◻. E. The series diverges because the limit used in the Ratio Test is not less than or equal to 1. F. The series converges conditionally because the limit used in the Ratio Test is ◻.
Consider the series ∑ n = 2 ∞ x n n(ln n) 3. (a) Find the series' radius and interval of convergence. (b) For what values of x does the series converge absolutely? (c) For what values of x does the series converge conditionally? (a) Find the interval of convergence. x Find the radius of convergence. R = ◻ (b) For what values of x does the series converge absolutely? x (c) For what values of x does the series converge conditionally? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The series converges conditionally at x = ◻. (Use a comma to separate answers as needed.) B. The series does not converge conditionally.