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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Section 12.1 Sigma Notation
a. The Symbol Σ
Section 12.2 Infinite Series
a. Partial Sums
b. Definition
c. Example
d. Geometric Series
e. Example
f. Theorem 12.2.4
g. Theorems 12.2.5
Section 12.3 The Integral Test; Basic Comparison, Limit Comparison
a. Theorem 12.3.1
b. The Integral Test
c. Harmonic Series and p-Series
d. Properties of Convergence and Divergence
e. Basic Comparison Theorem
f. Applying the Basic Comparison Theorem
g. Limit Comparison Theorem
h. Applying the Limit Comparison Theorem
Section 12.4 The Root Test; The Ratio Test
a. The Root Test
b. Applying the Root Test
c. The Ratio Test
d. Applying the Ratio Test
e. Summary on Convergence Tests
Chapter 12: Infinite Series
Section 12.5 Absolute Convergence and Conditional Convergence;
Alternating Series
a. Absolute Convergence and Conditional Convergence
b. Alternating Series
c. Estimating the Sum of an Alternating Series
Section 12.6 Taylor Polynomials in x; Taylor Series in x
a. Taylor’s Theorem
b. Corollary: Lagrange Formula for the Remainder
c. Taylor Series
d. Properties
Section 12.7 Taylor Polynomials and Taylor Series in x – a a. Taylor’s Theorem
b. nth Taylor Polynomial for g in powers of x – a
c. Taylor Expansion of g(x) in powers of x – a
Section 12.8 Power Series
a. Definition and Theorem
b. Case 1, 2 and 3
c. Radius of Convergence
Section 12.9 Differentiation & Integration of Power Series
a. Theorem 12.9.1
b. The Differentiability Theorem
c. Differentiating a Power Series
d. Term-by-Term Integration
e. Abel’s Theorem
f. Power Series: Taylor Series
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Sigma Notation
The symbol Σ is the capital Greek letter “sigma.” We write (1)
(“the sum of the a k from k equals 0 to k equals n”) to indicate the sum a 0 + a 1 + · · · + a n .
More generally, for n ≥ m, we write (2)
to indicate the sum
a m + a m+1 + · · · + a n .
In (1) and (2) the letter “k” is being used as a “dummy” variable; namely, it can be replaced by any other letter not already engaged. For instance,
all mean the same thing:
a 3 + a 4 + a 5 + a 6 + a 7 .
0 n
k k
a
∑ =
n k k m
a
∑ =
7 7 7
3 3 3
, ,
i j k
i j k
a a a
= = =
∑ ∑ ∑
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Infinite Series
Introduction; Definition
To form an infinite series, we begin with an infinite sequence of real numbers:
a 0 , a 1 , a 2 , . . . . We can’t form the sum of all the a k (there are an infinite number of them), but we can form the partial sums:
0
0 0
0 1
1 0 1
0 2
2 0 1 2
0 3
3 0 1 2 3
0
0 1 2 3
0
, ,
, ,
and so on.
k k
k k
k k
k k
n
n n k
k
s a a
s a a a
s a a a a
s a a a a a
s a a a a a a
=
=
=
=
=
= =
= + =
= + + =
= + + + =
= + + + + + =
∑
∑
∑
∑
∑
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Infinite Series
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Infinite Series
Example The series
illustrate two forms of divergence: bounded divergence, unbounded divergence.
For the first series,
s n = 1 − 1 + 1 − 1+ · · · +(−1) n . Here
Since s n > 2 n , the sum tends to ∞, and the series diverges. This is an example of unbounded divergence.
The sequence of partial sums reduces to 1, 0, 1, 0, . . . . Since the sequence diverges, the series diverges. This is an example of bounded divergence.
For the second series,
( )
0 0
1 k and 2 k
k k
∞ ∞
= =
∑ − ∑
1, if is even 0, if is odd
n
s n
n
=
2 0
2 1 2 2 2
n
k n
n k
s
=
= ∑ = + + + +
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Infinite Series
The Geometric Series The geometric progression
1, x, x 2 , x 3 , . . . gives rise to the numbers
1, 1 + x, 1 + x + x 2 , 1 + x + x 2 + x 3 , . . . .
These numbers are the partial sums of what is called the geometric series:
0 k k
x
∞
∑ =
This series is so important that we will give it special attention.
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Infinite Series
Setting x = ½ in (12.2.2), we have
By beginning the summation at k = 1 instead of at k = 0, we drop the term 1/2 0 = 1 and obtain
The partial sums of this series are given below and illustrated in Figure 12.2.1.
0 1
2
1 1
2 1 2
n k k =
= =
∑ −
1
2
3
4
5
1 2
1 1 3 2 4 4 1 1 1 7 2 4 8 8
1 1 1 1 15 2 4 8 16 16
1 1 1 1 1 31 2 4 8 16 32 32 s
s s s s
=
= + =
= + + =
= + + + =
= + + + + =
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Infinite Series
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Infinite Series
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The Integral Test; Basic Comparison, Limit Comparison
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
The Integral Test; Basic Comparison, Limit Comparison
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
The Integral Test; Basic Comparison, Limit Comparison
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
In the absence of detailed indexing, we cannot know definite limits, but we can be sure of the following:
The Integral Test; Basic Comparison, Limit Comparison
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The Integral Test; Basic Comparison, Limit Comparison
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Applying the Basic Comparison Theorem Example
(a) converges by comparison with
(b) converges by comparison with
3
1 2 k + 1
∑ ∑ k 1 3
3 3 3
1 1 1
and converges
2 k 1 < k k
+ ∑
3
5 4
5 7
k k + k +
∑ ∑ k 1 2
3 3
5 4 5 2 2
1 1
and converges
5 7
k k
k k < k = k k
+ + ∑
The Integral Test; Basic Comparison, Limit Comparison
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The Integral Test; Basic Comparison, Limit Comparison
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Applying the Limit Comparison Theorem Example
Determine whether the series converges or diverges.
Solution For large k
differs little from As k →∞,
Since
converges,
(it is a convergent geometric series), the original series converges.
1
1 5 k 3
k
∞
= −
∑
1 5 k − 3
1 5 k
1 5 k
∑
1 1 5 1
5 3 5 5 3 1 3 / 5 1
k
k ÷ k = k = k →
− − −
The Integral Test; Basic Comparison, Limit Comparison
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The Root Test; The Ratio Test
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
The Root Test; The Ratio Test
Example For the series
The series converges.
Applying the Root Test
( )
1 ln k k
∑
( ) 1/ 1 0
ln
k
a k
= k →
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The Root Test; The Ratio Test
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The Root Test; The Ratio Test
Applying the Ratio Test Example
For the series
10 k
∑ k
1
1
1 10 1 1 1
10 10 10
k k
k k
a k k
a k k
+
+
+ +
= ⋅ = →
The series converges.
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
The Root Test; The Ratio Test
Summary on Convergence Tests
In general, the root test is used only if powers are involved. The ratio test is particularly effective with factorials and with combinations of powers and factorials. If the terms are rational functions of k, the ratio test is inconclusive and the root test is difficult to apply. Series with rational terms are most easily handled by limit comparison with a p-series, a series of the form Σ 1/k p . If the terms have the configuration of a derivative, you may be able to apply the
integral test. Finally, keep in mind that, if a k 0, then there is no reason to try any convergence test; the series diverges.
→ /
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Absolute and Conditional Convergence
Absolute Convergence and Conditional Convergence
Series Σa k for which Σ|a k | converges are called absolutely convergent. The
theorem we have just proved says that
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Absolute and Conditional Convergence
in which consecutive terms have opposite signs is a called an alternating series.
As in our example, we shall follow custom and begin all alternating series with a positive term. In general, then, an alternating series will look like this:
Alternating Series A series such as
with all the a k positive. In this setup the partial sums of even index end with a positive term and the partial sums of odd index end with a negative term.
1 1 1 1 1
1 − + − + − + 2 3 4 5 6
( )
0 1 2 3
0
1 k k
k
a a a a a
∞
=
− + − + = ∑ −
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Absolute and Conditional Convergence
Estimating the Sum of an Alternating Series
You have seen that if a 0 , a 1 , a 2 , . . . is a decreasing sequence of positive numbers that tends to 0, then
converges to some sum L.
( )
0
1 k k
k
a
∞
=
∑ −
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Taylor Polynomials in x; Taylor Series in x
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Taylor Polynomials in x; Taylor Series in x
The following estimate for R n (x) is an immediate consequence of Corollary 12.6.2:
where J is the closed interval that joins 0 to x.
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Taylor Polynomials in x; Taylor Series in x
Taylor Series in x
By definition 0! = 1. Adopting the convention that f (0) = f , we can write Taylor polynomials
in Σ notation:
In this case, we say that f (x) can be expanded as a Taylor series in x and write
( ) ( ) ( ) ( ) 0 2 ( ) ( ) 0
0 0
2! !
n
n n
f f
P x f f x x x
n
′ ′′
= + + + +
( ) ( ) ( )
0
0 .
!
n k
k n
k
P x f x
= k
= ∑
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Taylor Polynomials in x; Taylor Series in x
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Taylor Polynomials and Taylor Series in x – a
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Taylor Polynomials and Taylor Series in x – a
The polynomial
is called the nth Taylor polynomial for g in powers of x − a. In this more general setting, the Lagrange formula for the remainder, R n (x), takes the form
where c is some number between a and x.
Now let x ∈ I, x ≠ a, and let J be the closed interval that joins a to x. Then
( ) ( ) ( )( ) ( )( ) 2 ( ) ( )( )
2! !
n
n n
g a g a
P x g a g a x a x a x a
n
′ ′′
= + − + − + + −
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Taylor Polynomials and Taylor Series in x – a
If R n (x ) → 0, then we have the series representation
which, in sigma notation, takes the form
( ) ( ) ( )( ) ( )( ) 2 ( ) ( )( ) ,
2! !
n
n n
g a g a
P x g a g a x a x a x a
n
′ ′′
= + − + − + + − +
This is known as the Taylor expansion of g(x) in powers of x − a. The series
on the right is called a Taylor series in x − a.
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Power Series
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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Power Series
There are exactly three possibilities for a power series:
Case 1. The series converges only at x = 0. This is what happens with
For x ≠ 0, k k x k 0, and so the series cannot converge.
Case 2. The series converges absolutely at all real numbers x. This is what happens with the exponential series
Case 3. There exists a positive number r such that the series converges absolutely for |x| < r and diverges for |x| > r . This is what happens with the geometric series
→ /
k k .
∑ k x
! . x k
∑ k
Here there is absolute convergence for |x| < 1 and divergence for |x| > 1.
k .
∑ x
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Power Series
Associated with each case is a radius of convergence:
In Case 1, we say that the radius of convergence is 0.
In Case 2, we say that the radius of convergence is ∞.
In Case 3, we say that the radius of convergence is r.
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Differentiation & Integration of Power Series
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Differentiation & Integration of Power Series
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Differentiation & Integration of Power Series
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Differentiation & Integration of Power Series
Term-by-term integration can be expressed by writing
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Differentiation & Integration of Power Series
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