1. 11.2 Series In this section, we will learn about: Various types of series. INFINITE SEQUENCES AND SERIES

2. SERIES If we try to add the terms of an infinite sequence we get an expression of the form a 1 + a 2 + a 3 + ··· + a n + ∙·∙ Series 1

3. INFINITE SERIES This is called an infinite series (or just a series).  It is denoted, for short, by the symbol

4. However, does it make sense to talk about the sum of infinitely many terms? INFINITE SERIES

5. It would be impossible to find a finite sum for the series 1 + 2 + 3 + 4 + 5 + ∙∙∙ + n + ···  If we start adding the terms, we get the cumulative sums 1, 3, 6, 10, 15, 21,...  After the nth term, we get n(n + 1)/2, which becomes very large as n increases. INFINITE SERIES

6. However, if we start to add the terms of the series we get: INFINITE SERIES

7. The table shows that, as we add more and more terms, these partial sums become closer and closer to 1.  In fact, by adding sufficiently many terms of the series, we can make the partial sums as close as we like to 1. INFINITE SERIES

8. So, it seems reasonable to say that the sum of this infinite series is 1 and to write: INFINITE SERIES

9. We use a similar idea to determine whether or not a general series (Series 1) has a sum. INFINITE SERIES

10. We consider the partial sums s 1 = a 1 s 2 = a 1 + a 2 s 3 = a 1 + a 2 + a 3 s 3 = a 1 + a 2 + a 3 + a 4  In general, INFINITE SERIES

11. These partial sums form a new sequence {s n }, which may or may not have a limit. If exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series  a n. INFINITE SERIES

12. Given a series let s n denote its nth partial sum: SUM OF INFINITE SERIES Definition 2

13. If the sequence {s n } is convergent and exists as a real number, then the series  a i is called convergent and we write:  The number s is called the sum of the series.  Otherwise, the series is called divergent. SUM OF INFINITE SERIES Definition 2

14. Thus, the sum of a series is the limit of the sequence of partial sums.  So, when we write, we mean that, by adding sufficiently many terms of the series, we can get as close as we like to the number s. SUM OF INFINITE SERIES

15. Notice that: SUM OF INFINITE SERIES

16. Compare with the improper integral  To find this integral, we integrate from 1 to t and then let t → .  For a series, we sum from 1 to n and then let n → . SUM OF INFINITE SERIES VS. IMPROPER INTEGRALS

17. An important example of an infinite series is the geometric series GEOMETRIC SERIES Example 1

18. Each term is obtained from the preceding one by multiplying it by the common ratio r.  We have already considered the special case where a = ½ and r = ½ earlier in the section. GEOMETRIC SERIES Example 1

19. If r = 1, then s n = a + a + ∙∙∙ + a = na →   Since does not exist, the geometric series diverges in this case. GEOMETRIC SERIES Example 1

20. If r  1, we have s n = a + ar + ar 2 + ∙∙∙ + ar n–1 and rs n = ar + ar 2 + ∙∙∙ +ar n–1 + ar n GEOMETRIC SERIES Example 1

21. Subtracting these equations, we get: s n – rs n = a – ar n and GEOMETRIC SERIES E. g. 1—Equation 3

22. If –1 < r < 1, we know from Result 9 in Section 11.1 that r n → 0 as n → . So,  Thus, when |r | < 1, the series is convergent and its sum is a/(1 – r). GEOMETRIC SERIES Example 1

23. If r  –1 or r > 1, the sequence {r n } is divergent by Result 9 in Section 11.1 So, by Equation 3, does not exist.  Hence, the series diverges in those cases. GEOMETRIC SERIES Example 1

24. The figure provides a geometric demonstration of the result in Example 1. GEOMETRIC SERIES

25. If s is the sum of the series, then, by similar triangles, So, GEOMETRIC SERIES

26. We summarize the results of Example 1 as follows. The geometric series is convergent if |r | < 1. Result 4

27. Moreover, the sum of the series is: If |r |  1, the series is divergent. GEOMETRIC SERIES Result 4

28. Find the sum of the geometric series  The first term is a = 5 and the common ratio is r = –2/3 GEOMETRIC SERIES Example 2

29. Since |r | = 2/3 < 1, the series is convergent by Result 4 and its sum is: GEOMETRIC SERIES Example 2

30. What do we really mean when we say that the sum of the series in Example 2 is 3?  Of course, we cannot literally add an infinite number of terms, one by one. GEOMETRIC SERIES

31. However, according to Definition 2, the total sum is the limit of the sequence of partial sums.  So, by taking the sum of sufficiently many terms, we can get as close as we like to the number 3. GEOMETRIC SERIES

32. The table shows the first ten partial sums s n. The graph shows how the sequence of partial sums approaches 3. GEOMETRIC SERIES

33. Is the series convergent or divergent? GEOMETRIC SERIES Example 3

34. Let us rewrite the nth term of the series in the form ar n-1 :  We recognize this series as a geometric series with a = 4 and r = 4/3.  Since r > 1, the series diverges by Result 4. GEOMETRIC SERIES Example 3

35. Write the number as a ratio of integers.   After the first term, we have a geometric series with a = 17/10 3 and r = 1/10 2. GEOMETRIC SERIES Example 4

36. Therefore, GEOMETRIC SERIES Example 4

37. Find the sum of the series where |x| < 1.  Notice that this series starts with n = 0.  So, the first term is x 0 = 1.  With series, we adopt the convention that x 0 = 1 even when x = 0. GEOMETRIC SERIES Example 5

38. Thus,  This is a geometric series with a = 1 and r = x. GEOMETRIC SERIES Example 5

39. Since |r | = |x| < 1, it converges, and Result 4 gives: GEOMETRIC SERIES E. g. 5—Equation 5

40. Show that the series is convergent, and find its sum. SERIES Example 6

41. This is not a geometric series.  So, we go back to the definition of a convergent series and compute the partial sums: SERIES Example 6

42. We can simplify this expression if we use the partial fraction decomposition.  See Section 7.4 SERIES Example 6

43. Thus, we have: SERIES Example 6

44. Thus,  Hence, the given series is convergent and SERIES Example 6

45. The figure illustrates Example 6 by showing the graphs of the sequence of terms a n =1/[n(n + 1)] and the sequence {s n } of partial sums.  Notice that a n → 0 and s n → 1. SERIES

46. Show that the harmonic series is divergent. HARMONIC SERIES Example 7

47. For this particular series it’s convenient to consider the partial sums s 2, s 4, s 8, s 16, s 32, …and show that they become large. HARMONIC SERIES Example 7

48. HARMONIC SERIES Example 7 Similarly,

49. HARMONIC SERIES Example 7 Similarly,

50. Similarly, s 32 > 1 + 5/2, s 64 > 1 + 6/2, and, in general,  This shows that s 2 n →  as n → , and so {s n } is divergent.  Therefore, the harmonic series diverges. HARMONIC SERIES Example 7

51. HARMONIC SERIES The method used in Example 7 for showing that the harmonic series diverges is due to the French scholar Nicole Oresme (1323–1382).

52. If the series is convergent, then SERIES Theorem 6

53. Let s n = a 1 + a 2 + ∙∙∙ + a n Then, a n = s n – s n–1  Since  a n is convergent, the sequence {s n } is convergent. SERIES Theorem 6 - Proof

54. Let Since n – 1 →  as n → , we also have: SERIES Theorem 6 - Proof

55. Therefore, SERIES Theorem 6 - Proof

56. With any series  a n we associate two sequences:  The sequence {s n } of its partial sums  The sequence {a n } of its terms SERIES Note 1

57. If  a n is convergent, then  The limit of the sequence {s n } is s (the sum of the series).  The limit of the sequence {a n }, as Theorem 6 asserts, is 0. SERIES Note 1

58. The converse of Theorem 6 is not true in general.  If, we cannot conclude that  a n is convergent. SERIES Note 2

59. Observe that, for the harmonic series  1/n, we have a n = 1/n → 0 as n → ∞.  However, we showed in Example 7 that  1/n is divergent. SERIES Note 2

60. If does not exist or if, then the series is divergent. THE TEST FOR DIVERGENCE Test 7

61. The Test for Divergence follows from Theorem 6.  If the series is not divergent, then it is convergent.  Thus, TEST FOR DIVERGENCE

62. Show that the series diverges.   So, the series diverges by the Test for Divergence. TEST FOR DIVERGENCE Example 8

63. If we find that, we know that  a n is divergent. If we find that, we know nothing about the convergence or divergence of  a n. SERIES Note 3

64. Remember the warning in Note 2:  If, the series  a n might converge or diverge. SERIES Note 3

65. If  a n and  b n are convergent series, then so are the series  ca n (where c is a constant),  (a n + b n ), and  (a n – b n ), and SERIES Theorem 8

66. These properties of convergent series follow from the corresponding Limit Laws for Sequences in Section 11.1  For instance, we prove part ii of Theorem 8 as follows. SERIES

67. Let THEOREM 8 ii—PROOF

68. The nth partial sum for the series Σ (a n + b n ) is: THEOREM 8 ii—PROOF

69. Using Equation 10 in Section 5.2, we have: THEOREM 8 ii—PROOF

70. Hence, Σ (a n + b n ) is convergent, and its sum is: THEOREM 8 ii—PROOF

71. Find the sum of the series  The series  1/2 n is a geometric series with a = ½ and r = ½.  Hence, SERIES Example 9

72. In Example 6, we found that: So, by Theorem 8, the given series is convergent and SERIES Example 9

73. A finite number of terms does not affect the convergence or divergence of a series. SERIES Note 4

74. For instance, suppose that we were able to show that the series is convergent.  Since it follows that the entire series is convergent. SERIES Note 4

75. Similarly, if it is known that the series converges, then the full series is also convergent. SERIES Note 4