infinite series
DESCRIPTION
Infinite Series. Objective: We will try to find the sum of a series with infinitely many terms. Sums of Infinite Series. Our first objective is to define what is meant by the “sum” of infinitely many real numbers. We start with a definition:. Sums of Infinite Series. - PowerPoint PPT PresentationTRANSCRIPT
Infinite Series
Objective: We will try to find the sum of a series with infinitely many
terms.
Sums of Infinite Series
• Our first objective is to define what is meant by the “sum” of infinitely many real numbers. We start with a definition:
Sums of Infinite Series
• Our first objective is to define what is meant by the “sum” of infinitely many real numbers. We start with a definition:
Sums of Infinite Series
• Since it is impossible to add infinitely many numbers directly, sums of infinite series are defined and computed by an indirect limiting process. We will consider the decimal 0.3333… This can be viewed as the infinite series 0.3 + 0.03 + 0.003 + … or as
• Since this is the decimal expansion of 1/3, any reasonable definition for the sum of this series should yield 1/3.
...10
3
10
3
10
332
Sums of Infinite Series
• To obtain such a definition, consider the following sequence of (finite) sums.
3.010
31 s
33.010
3
10
322 s
333.010
3
10
3
10
3323 s
3333.010
3
10
3
10
3
10
34324 s
Sums of Infinite Series
• The sequence of numbers s1 , s2 , s3 , s4 , … can be viewed as a succession of approximations to the “sum” of the infinite series, which we want to be 1/3. As we progress through the sequence, more and more terms of the infinite series are used, and the approximations get better and better, suggesting that the desired sum of 1/3 might be the limit. We need to calculate the limit of the general term in the sequence, namely
nns 10
3...
10
3
10
32
Closed Form
• In formulas such as
• The left side is said to express the sum in open form and the right side is said to express the sum in closed form. The open form indicates the summands and the closed form is an explicit formula for the sum. In other words, in closed form, if you know the value of n, you know the sum.
n
k
k1 2
)1(...21
nnn
Closed Form
• Express in closed form.
n
k
k1
2)3(
Closed Form
• Express in closed form.
n
k
n
k
n
k
n
k
kkk1 1 1
22
1
69)3(
n
k
k1
2)3(
Closed Form
• Express in closed form.
n
k
n
k
n
k
n
k
kkk1 1 1
22
1
69)3(
6
)12)(1(
2
)1(69
nnnnnn
6
73212
6
32339
23232 nnnnnn
nnn
n
k
k1
2)3(
Sums of Infinite Series
• The problem with calculating is complicated by the fact that both the last term
and the number of terms change with n. It is best to rewrite such limits in a closed form in which the number of terms does not vary, if possible. To do this, we multiply both sides of the equation for the general term by 1/10 to obtain:
nnn
ns
10
3...
10
3
10
3limlim
2
132 10
3...
10
3
10
3
10
1nns
Sums of Infinite Series
• Now subtract the new equation from the original.
132 10
3...
10
3
10
3
10
1nns
nns 10
3...
10
3
10
32
Sums of Infinite Series
• Subtracting the new equation from the original gives
132 10
3...
10
3
10
3
10
1nns
nns 10
3...
10
3
10
32
110
3
10
3
10
9nns
nns 10
11
10
3
10
9
Sums of Infinite Series
• The general term can now be written as
110
3
10
3
10
9nns
nns 10
11
10
3
10
9
nns 10
11
3
1
Sums of Infinite Series
• Now, the limit becomes
3
1
10
11
3
1limlim
nnn
ns
nns 10
11
3
1
Sums of Infinite Series
• Motivated by the previous example, we are now ready to define the general concept of the “sum” of an infinite series .......321 kuuuu
Sums of Infinite Series
• Motivated by the previous example, we are now ready to define the general concept of the “sum” of an infinite series .
• We begin with some terminology. Let sn denote the sum of the initial terms of the series, up to and including the term with index n. Thus,
......321 kuuuu
11 us
212 uus
3213 uuus
43214 uuuus
n
kknn uuuuus
1321 ...
Sums of Infinite Series
• The number sn is called the nth partial sum of the series and the sequence is called the sequence of partial sums. As n increases, the partial sum sn = u1 + u2 + …+ un + … includes more and more terms of the series. Thus, if sn tends toward a limit as
, it is reasonable to view this limit as the sum of all the terms in the series.
1nns
n
Sums of Infinite Series
• This suggests the following definition:
Example 1
• Determine whether the series 1 – 1 + 1 – 1 + 1- 1 +… converges or diverges. If it converges, find the sum.
Example 1
• Determine whether the series 1 – 1 + 1 – 1 + 1- 1 +… converges or diverges. If it converges, find the sum.• It is tempting to conclude that the sum of the series is
zero by arguing that the positive and negative terms will cancel. This is not correct. We will look at the partial sums.
Example 1
• Determine whether the series 1 – 1 + 1 – 1 + 1- 1 +… converges or diverges. If it converges, find the sum.• It is tempting to conclude that the sum of the series is
zero by arguing that the positive and negative term will cancel. This is not correct. We will look at the partial sums.
• Thus, the sequence of partial sums is 1, 0, 1, 0, 1, … This is a divergent series and has no sum.
11 s 0112 s 11113 s
Geometric Series
• In many important series, each term is obtained by multiplying the preceding term by some fixed constant. Thus, if the initial term of the series is a and each term is obtained by multiplying the preceding term by r, then the series has the form
)0...(...2
0
aarararaar k
k
k
Geometric Series
• In many important series, each term is obtained by multiplying the preceding term by some fixed constant. Thus, if the initial term of the series is a and each term is obtained by multiplying the preceding term by r, then the series has the form
• Such series are called geometric series, and the number r is called the ratio for the series.
)0...(...2
0
aarararaar k
k
k
Geometric Series
• Here are some examples of geometric series.
Geometric Series
• The following theorem is the fundamental result on convergence or geometric series.
Example 2
• The series
is a geometric series with a = 5 and r = ¼. Since |r|=1/4 < 1, the series converges and the sum is
...4
5...
4
5
4
55
4
52
0
k
kk
3
20
4/11
5
1
ra
Example 3
• Find the rational number represented by the repeating decimal 0.784784784….
Example 3
• Find the rational number represented by the repeating decimal 0.784784784….
• We can write this as 0.784 + 0.000784 + 0.000000784 so the given decimal is the sum of a geometric series with a = 0.784 and r = 0.001. Thus, the sum is
999
784
999.
784.
001.01
784.0
1
ra
Example 4
• In each part, determine whether the series converges, and if so find its sum.
1
12 53)(k
kka
0
)(k
kxb
Example 4
• In each part, determine whether the series converges, and if so find its sum.
• (a) This is a geometric series in a concealed form, since we can rewrite it as
1
12 53)(k
kka
0
)(k
kxb
1
111
1
12
5
99
5
953)(
k
kkk
k
k
kka divergesr ;15
9
Example 4
• In each part, determine whether the series converges, and if so find its sum.
• (b) This is a geometric series with a = 1 and r = x, so it converges if |x| < 1 and diverges otherwise. When it converges its sum is
1
12 53)(k
kka
0
)(k
kxb
xx
k
k
1
1
0
Telescoping Sums
• Determine whether the series converges or diverges.• If it converges, find the sum.
...43
1
32
1
21
1
)1(
1
1
k kk
Telescoping Sums
• Determine whether the series converges or diverges.• If it converges, find the sum.• To write this in closed form, we will use the method
of partial fractions to obtain
...43
1
32
1
21
1
)1(
1
1
k kk
1
11
)1(
1
1
kkkkk
Telescoping Sums
• Determine whether the series converges or diverges.• If it converges, find the sum.
...43
1
32
1
21
1
)1(
1
1
k kk
n
kn kks
1 1
11
1
11
4
1
3
1
3
1
2
1
2
11
nn
Telescoping Sums
• Determine whether the series converges or diverges.• If it converges, find the sum.
...43
1
32
1
21
1
)1(
1
1
k kk
n
kn kks
1 1
11
1
11
4
1
3
1
3
1
2
1
2
11
nn
11
11limlim
)1(
1
1
n
skk n
nn
k
Harmonic Series
• One of the most important of all diverging series is the harmonic series
which arises in connection with the overtones produced by a vibrating musical string. It is not immediately evident that this series diverges. However, the divergence will become apparent when we examine the partial sums in detail.
1
1
k k
Harmonic Series
• Because the terms in the series are all positive, the partial sums
form a strictly increasing sequence. Thus, by Theorem 10.2.3 there is no constant M that is greater than or equal to every partial sum, so the series diverges.
11 s2
112 s
3
1
2
113 s
4
1
3
1
2
114 s
Homework
• Pages 649-650• 1-19 odd