Chapter 12: Infinite Series

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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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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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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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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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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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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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The Integral Test; Basic Comparison, Limit Comparison

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The Integral Test; Basic Comparison, Limit Comparison

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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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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

+ + ∑

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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

− − −

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The Root Test; The Ratio Test

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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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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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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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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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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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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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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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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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Differentiation & Integration of Power Series

Power Series; Taylor Series It is time to relate Taylor series

to power series in general. The relationship is very simple.

( ) ( )

0

0

!

k

k k

f x

k

∑ =

Figure

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