INFINITE SEQUENCES AND SERIES

1
11
INFINITE SEQUENCES AND SERIES
2
INFINITE SEQUENCES AND SERIES
11.2Series
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
INFINITE SERIES

• 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
INFINITE SERIES

• 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.

7
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
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 S an.

14
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 S
  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 ? 8.
• For a series, we sum from 1 to n and then let n ?
  8.

19
GEOMETRIC SERIES
Example 1

• 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
GEOMETRIC SERIES
Example 1

• If r 1, then
• sn a a a na ? 8
• Since doesnt exist, the geometric
  series diverges in this case.

22
GEOMETRIC SERIES
Example 1

• If r ? 1, we have
• sn a ar ar2 ar n1
• and
• rsn ar ar2 ar n1 ar n

23
GEOMETRIC SERIES
E. g. 1Equation 3

• Subtracting these equations, we get
• sn rsn a ar n

24
GEOMETRIC SERIES
Example 1

• If 1 lt r lt 1, we know from Result 9 in Section
  11.1 that r n ? 0 as n ? 8.
• So,
• Thus, when r lt 1, the series is convergent
  and its sum is a/(1 r).

25
GEOMETRIC SERIES
Example 1

• If r 1 or r gt 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
GEOMETRIC SERIES

• If s is the sum of the series, then, by similar
  triangles,
• So,

28
GEOMETRIC SERIES

• We summarize the results of Example 1 as follows.

29
GEOMETRIC SERIES
Result 4

• The geometric series
• is convergent if r lt 1.

30
GEOMETRIC SERIES
Result 4

• The sum of the series is
• If r 1, the series is divergent.

31
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 lt 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 cant 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
GEOMETRIC SERIES

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

36
GEOMETRIC SERIES
Example 3

• Is the series convergent or divergent?

37
GEOMETRIC SERIES
Example 3

• Lets 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 gt 1, the series diverges by Result 4.

38
GEOMETRIC SERIES
Example 4

• Write the number as a ratio of integers.
• 2.3171717
• After the first term, we have a geometric series
  with a 17/103 and r 1/102.

39
GEOMETRIC SERIES
Example 4

• Therefore,

40
GEOMETRIC SERIES
Example 5

• Find the sum of the series where x lt 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
GEOMETRIC SERIES
Example 5

• Thus,
• This is a geometric series with a 1 and r x.

42
GEOMETRIC SERIES
E. g. 5Equation 5

• Since r x lt 1, it converges, and Result 4
  gives

43
SERIES
Example 6

• Show that the series
• is convergent, and find its sum.

44
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 convergentand

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
HARMONIC SERIES
Example 7

• Show that the harmonic series
• is divergent.

50
HARMONIC SERIES
Example 7

• For this particular series its 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
HARMONIC SERIES
Example 7

• Similarly, s32 gt 1 5/2, s64 gt 1 6/2, and, in
  general,
• This shows that s2n ? 8 as n ? 8, 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 (13231382).

55
SERIES
Theorem 6

• If the series is convergent, then

56
SERIES
Theorem 6Proof

• Let sn a1 a2 an
• Then, an sn sn1
• Since S an is convergent, the sequence sn is
  convergent.

57
SERIES
Theorem 6Proof

• Let
• Since n 1 ? 8 as n ? 8, we also have

58
SERIES
Theorem 6Proof

• Therefore,

59
SERIES
Note 1

• With any series S an we associate two sequences
  
• The sequence sn of its partial sums
• The sequence an of its terms

60
SERIES
Note 1

• If S 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
SERIES
Note 2

• The converse of Theorem 6 is not true in
  general.
• If , we cannot conclude that S
  an is convergent.

62
SERIES
Note 2

• Observe that, for the harmonic series S 1/n, we
  have an 1/n ? 0 as n ? 8.
• However, we showed in Example 7 that S 1/n is
  divergent.

63
THE TEST FOR DIVERGENCE
Test 7

• If does not exist or if
  , then the series
• is divergent.

64
TEST FOR DIVERGENCE

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

65
TEST FOR DIVERGENCE
Example 8

• Show that the series diverges.
• So, the series diverges by the Test for
  Divergence.

66
SERIES
Note 3

• If we find that , we know that S
  an is divergent.
• If we find that , we know
  nothing about the convergence or divergence of S
  an.

67
SERIES
Note 3

• Remember the warning in Note 2
• If , the series S an might
  converge or diverge.

68
SERIES
Theorem 8

• If S an and S bn are convergent series, then so
  are the series S can (where c is a constant), S
  (an bn), and S (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 iiPROOF

• Let

71
THEOREM 8 iiPROOF

• The nth partial sum for the series S (an bn)
  is

72
THEOREM 8 iiPROOF

• Using Equation 10 in Section 5.2, we have

73
THEOREM 8 iiPROOF

• Hence, S (an bn) is convergent, and its sum is

74
SERIES
Example 9

• Find the sum of the series
• The series S 1/2n is a geometric series with a
  ½ and r ½.
• Hence,

75
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 doesnt 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.