Theory and ExamplesThe sign of the remainder of an

Problem 1E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 3E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 70E Theory and Examples Outline of the proof of the Rearrangement Theorem (Theorem 17) a. Let be a positive real number, let and let Show that for some index N1 and for some index N2? N1 Since all the terms a1, a2 , …… , aN2 appear somewhere in the sequence {bn}, there is an index N3? N2 such that if n ? N3 then is at most a sum of terms with m ? N1Therefore, if n ? N3. b. The argument in part (a) shows that if converges absolutely then converges and Now show that because , converges to

Problem 2E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 4E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 6E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 5E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 7E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 8E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 9E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 13E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 10E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 11E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 14E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 12E Determining Convergence or Divergence In Exercise, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

Problem 15E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 17E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 18E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 16E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 19E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 20E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 21E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 22E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 23E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 24E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 25E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 27E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 28E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 26E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 29E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 30E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 31E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 32E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 33E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 35E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 34E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 37E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 36E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 39E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 38E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 41E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 40E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 42E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 43E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 44E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 45E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 46E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 47E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 48E Absolute and Conditional Convergence Which of the series in Exercise converge absolutely, which converge, and which diverge? Give reasons for your answers.

Problem 49E Error Estimation In Exercise, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.

Problem 50E Error Estimation In Exercise, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.

Problem 51E Error Estimation In Exercise, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.

Problem 52E Error Estimation In Exercise, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.

Problem 53E Error Estimation In Exercise, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.

Problem 54E Error Estimation In Exercise, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.

Problem 55E Error Estimation In Exercise, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.

Problem 56E Error Estimation In Exercise, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.

Problem 58E Error Estimation Approximate the sums with an error of magnitude less than .

Problem 57E Error Estimation Approximate the sums with an error of magnitude less than

Problem 59E Theory and Examples a. The series does not meet one of the conditions of Theorem 14. Which one? b. Use Theorem 17 to find the sum of the series in part (a). Reference: THEOREM 14— The Alternating Series Test (Leibniz’s Test) The series converges if all three of the following conditions are satisfied: 1. The un’s are all positive. 2. The positive un’s are (eventually) nonincreasing: for some integer N. 3. THEOREM 17— The Rearrangement Theorem for Absolutely Convergent Series If converges absolutely, and b1, b2 , ….., bn….. , is any arrangement of the sequence {an} then converges absolutely and

Problem 60E Theory and Examples The limit L of an alternating series that satisfies the conditions of Theorem 14 lies between the values of any two consecutive partial sums. This suggests using the average to estimate L. Compute as an approximation to the sum of the alternating harmonic series. The exact sum is ln 2 = 0.69314718……. Reference: THEOREM 14— The Alternating Series Test (Leibniz’s Test) The series converges if all three of the following conditions are satisfied: 1. The un’s are all positive. 2. The positive un’s are (eventually) non-increasing: for some integer N. 3.

Problem 61E Theory and Examples The sign of the remainder of an alternating series that satisfies the conditions of Theorem 14 Prove the assertion in Theorem 15 that whenever an alternating series satisfying the conditions of Theorem 14 is approximated with one of its partial sums, then the remainder (sum of the unused terms) has the same sign as the first unused term. (Hint: Group the remainder’s terms in consecutive pairs.) Reference: THEOREM 15— The Alternating Series Estimation Theorem If the alternating series satisfies the three conditions of Theorem 14, then for n ? N approximates the sum L of the series with an error whose absolute value is less than un+1 the absolute value of the first unused term. Furthermore, the sum L lies between any two successive partial sums sn and sn+1 and the remainder, L - sn , has the same sign as the first unused term. THEOREM 14— The Alternating Series Test (Leibniz’s Test) The series converges if all three of the following conditions are satisfied: 1. The un’s are all positive. 2. The positive un’s are (eventually) non-increasing: for some integer N. 3.

Problem 62E Theory and Examples Show that the sum of the first 2n terms of the series is the same as the sum of the first n terms of the series Do these series converge? What is the sum of the first 2n + 1 terms of the first series? If the series converge, what is their sum?

Problem 63E Theory and Examples Show that if diverges, then diverges.

Problem 64E Theory and Examples Show that if converges absolutely, then

Problem 65E Theory and Examples Show that if both converge absolutely, then so do the following.

Problem 66E Theory and Examples Show by example that may diverge even if and both converge.

Problem 67E If converges absolutely, prove that converges.

Problem 68E Does the series Converges or diverges? Justify your answer

Problem 69E Theory and Examples In the alternating harmonic series, suppose the goal is to arrange the terms to get a new series that converges to –1/2.Start the new arrangement with the first negative term, which is –1/2. Whenever you have a sum that is less than or equal to –1/2, start introducing positive terms, taken in order, until the new total is greater than –1/2.Then add negative terms until the total is less than or equal to –1/2 again. Continue this process until your partial sums have been above the target at least three times and finish at or below it. If sn is the sum of the first n terms of your new series, plot the points (n, sn) to illustrate how the sums are behaving.