1. INFINITE SEQUENCES AND SERIES11
INFINITE SEQUENCES AND SERIES

2. 11.2 Series In this section, we will learn about:
INFINITE SEQUENCES AND SERIES
11.2 Series
In this section, we will learn about:
Various types of series.

3. SERIES
Series 1
If we try to add the terms of an infinite sequence we get an expression of the form
a1 + a2 + a3 + ··· + an + ∙·∙

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

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

6. It would be impossible to find a finite sum for the series
INFINITE SERIES
It would be impossible to find a finite sum for the series
∙∙∙ + 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.

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

8. INFINITE SERIES
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.

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

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

11. We consider the partial sums s1 = a1 s2 = a1 + a2 s3 = a1 + a2 + a3
INFINITE SERIES
We consider the partial sums
s1 = a1
s2 = a1 + a2
s3 = a1 + a2 + a3
s3 = a1 + a2 + a3 + a4
In general,

12. INFINITE SERIES
These partial sums form a new sequence {sn}, which may or may not have a limit.

13. SUM OF INFINITE SERIES
If exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series Σ an.

14. let sn denote its nth partial sum:
SUM OF INFINITE SERIES
Definition 2
Given a series
let sn denote its nth partial sum:

15. SUM OF INFINITE SERIES
Definition 2
If the sequence {sn} is convergent and exists as a real number, then the series Σ an is called convergent and we write:
The number s is called the sum of the series.
Otherwise, the series is called divergent.

16. SUM OF INFINITE SERIES
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.

17. SUM OF INFINITE SERIES
Notice that:

18. SUM OF INFINITE SERIES VS. IMPROPER INTEGRALS
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 → ∞.

19. An important example of an infinite series is the geometric series

20. GEOMETRIC SERIES
Example 1
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.

21. If r = 1, then sn = a + a + ∙∙∙ + a = na → ±∞
GEOMETRIC SERIES
Example 1
If r = 1, then
sn = a + a + ∙∙∙ + a = na → ±∞
Since doesn’t exist, the geometric series diverges in this case.

22. rsn = ar + ar2 + ∙∙∙ +ar n–1 + ar n
GEOMETRIC SERIES
Example 1
If r ≠ 1, we have
sn = a + ar + ar2 + ∙∙∙ + ar n–1
and
rsn = ar + ar2 + ∙∙∙ +ar n–1 + ar n

23. Subtracting these equations, we get:
GEOMETRIC SERIES
E. g. 1—Equation 3
Subtracting these equations, we get:
sn – rsn = a – ar n

24. GEOMETRIC SERIES
Example 1
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).

25. So, by Equation 3, does not exist.
GEOMETRIC SERIES
Example 1
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.

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

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

28. We summarize the results of Example 1 as follows.
GEOMETRIC SERIES
We summarize the results of Example 1 as follows.

29. is convergent if |r | < 1.
GEOMETRIC SERIES
Result 4
The geometric series
is convergent if |r | < 1.

30. The sum of the series is:
GEOMETRIC SERIES
Result 4
The sum of the series is:
If |r | ≥ 1, the series is divergent.

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

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

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

34. GEOMETRIC SERIES
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.

35. The table shows the first ten partial sums sn.
GEOMETRIC SERIES
The table shows the first ten partial sums sn.
The graph shows how the sequence of partial sums approaches 3.

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

37. Let’s rewrite the nth term of the series in the form ar n-1:
GEOMETRIC SERIES
Example 3
Let’s 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.

38. Write the number as a ratio of integers.
GEOMETRIC SERIES
Example 4
Write the number as a ratio of integers. …
After the first term, we have a geometric series with a = 17/103 and r = 1/102.

39. GEOMETRIC SERIES
Example 4
Therefore,

40. Find the sum of the series where |x| < 1.
GEOMETRIC SERIES
Example 5
Find the sum of the series where |x| < 1.
Notice that this series starts with n = 0.
So, the first term is x0 = 1.
With series, we adopt the convention that x0 = 1 even when x = 0.

41. Thus, This is a geometric series with a = 1 and r = x.
Example 5
Thus,
This is a geometric series with a = 1 and r = x.

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

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

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

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

46. SERIES
Example 6
Thus, we have:

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

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

49. Show that the harmonic series
Example 7
Show that the harmonic series
is divergent.

50. HARMONIC SERIES
Example 7
For this particular series it’s convenient to consider the partial sums s2, s4, s8, s16, s32, … and show that they become large.

51. HARMONIC SERIES
Example 7
Similarly,

52. HARMONIC SERIES
Example 7
Similarly,

53. Similarly, s32 > 1 + 5/2, s64 > 1 + 6/2, and, in general,
HARMONIC SERIES
Example 7
Similarly, s32 > 1 + 5/2, s64 > 1 + 6/2, and, in general,
This shows that s2n → ∞ as n → ∞, and so {sn} is divergent.
Therefore, the harmonic series diverges.

54. 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).

55. If the series is convergent, then
Theorem 6
If the series is convergent, then

56. Let sn = a1 + a2 + ∙∙∙ + an Then, an = sn – sn–1
SERIES
Theorem 6—Proof
Let sn = a1 + a2 + ∙∙∙ + an
Then, an = sn – sn–1
Since Σ an is convergent, the sequence {sn} is convergent.

57. Since n – 1 → ∞ as n → ∞, we also have:
SERIES
Theorem 6—Proof
Let
Since n – 1 → ∞ as n → ∞, we also have:

58. SERIES
Theorem 6—Proof
Therefore,

59. With any series Σ an we associate two sequences:
Note 1
With any series Σ an we associate two sequences:
The sequence {sn} of its partial sums
The sequence {an} of its terms

60. If Σ an is convergent, then
SERIES
Note 1
If Σ an is convergent, then
The limit of the sequence {sn} is s (the sum of the series).
The limit of the sequence {an}, as Theorem 6 asserts, is 0.

61. The converse of Theorem 6 is not true in general.
SERIES
Note 2
The converse of Theorem 6 is not true in general.
If , we cannot conclude that Σ an is convergent.

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

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

64. The Test for Divergence follows from Theorem 6.
If the series is not divergent, then it is convergent.
Thus,

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

66. If we find that , we know that Σ an is divergent.
SERIES
Note 3
If we find that , we know that Σ an is divergent.
If we find that , we know nothing about the convergence or divergence of Σ an.

67. Remember the warning in Note 2:
SERIES
Note 3
Remember the warning in Note 2:
If , the series Σ an might converge or diverge.

68. SERIES
Theorem 8
If Σ an and Σ bn are convergent series, then so are the series Σ can (where c is a constant), Σ (an + bn), and Σ (an – bn), and

69. SERIES
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.

70. THEOREM 8 ii—PROOF
Let

71. The nth partial sum for the series Σ (an + bn) is:
THEOREM 8 ii—PROOF
The nth partial sum for the series Σ (an + bn) is:

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

73. Hence, Σ (an + bn) is convergent, and its sum is:
THEOREM 8 ii—PROOF
Hence, Σ (an + bn) is convergent, and its sum is:

74. Find the sum of the series
Example 9
Find the sum of the series
The series Σ 1/2n is a geometric series with a = ½ and r = ½.
Hence,

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

76. SERIES
Note 4
A finite number of terms doesn’t affect the convergence or divergence of a series.

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

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