Need help with 12 and 20

160' rystal Creek Mappin Project Data: Field Studies 021 810 CHAPTER 11 Sequences, Series, and Power Series order of the terms. For instance, a rearrangement of 2 an could start as follows: di + a2 + as + a3 + a4 + dis + as + a7 + azo t ... It turns out that if Ean is an absolutely convergent series with sum s, then any rearrangement of Z a, has the same sum s. However, any conditionally convergent series can be rearranged to give a different sum, To illustrate this fact let's consider the alternating harmonic series from Example 1. In Exercise 50 you are asked to show that 4 1 - 2 +3 - 4+ 3- 6+7- 8+ ... =In2 If we multiply this series by 2, we get 2 - 4+6 - 8+...=In2 Adding these zeros does not affect Inserting zeros between the terms of this series, we have the sum of the series; each term in the sequence of partial sums is 5 0 + z + 0 - 4 + 0 + 6 + 0- g + ... =In2 repeated, but the limit is the same. Now we add the series in Equations 4 and 5 using Theorem 1 1.2.8: 6 1 + 3 - 2 + 3 + 4 - 4+ ... = 3In2 Notice that the series in (6) contains the same terms as in (4) but rearranged so that one negative term occurs after each pair of positive terms. The sums of these series, however, are different. In fact, Riemann proved that if Ean is a conditionally convergent series and r is any real number what- soever, then there is a rearrangement of E a, that has a sum equal to r. A proof of this fact is outlined in Exercise 52. 11.5 Exercises 1. (a) What is an alternating series? (b) Under what conditions does an alternating series 9. E (- 1 )"en Vn converge? 1= 1 10. E (-1)"- 2 n + 3 (c) If these conditions are satisfied, what can you say about the remainder after n terms? 11. E (-1 ) 2+1 - n 2 1 = 1 n' + 4 12. E (-1)" - 2-20 Test the series for convergence or divergence. 2. 3-3+3-3+ 1-... 13. E (-1)"-12 14. E (-1)"-arctan n 3. -3+8- 9+8-"+... 4. 1 1 1 1 1 15. > sin(n + ) T 1 Cos ITT In 3 In 4 In 5 In 6 . . . In 7 "-0 1 + vn 16. 2 5. 5 (1) "-1 n=1 3 + 5n 2 17. > (-1)" sin 1=0 Vn + 1 1 = 1 n 18. 7. Z (-1)"- 3n - 1 2n + 1 8. (-1)- n' 19. E (-1 ) " n2 1 =1 n' + n + 1 20. [ (- 1)" ( Vn+ 1-Va)