### Main Menu

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

### Main Menu

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} *.*

_{k}

_{n}

*More generally, for n * *≥ m, we write* (2)

### to indicate the sum

*a* _{m} *+ a* _{m+1} *+ · · · + a* _{n} *.*

_{m}

_{m+1}

_{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*

### = = =

### ∑ ∑ ∑

### Main Menu

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:*

_{k}

### 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*

### =

### =

### =

### =

### =

### = =

### = + =

### = + + =

### = + + + =

### = + + + + + =

### ∑

### ∑

### ∑

### ∑

### ∑

###

### Main Menu

## Infinite Series

### Main Menu

## 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

_{n}

^{n}

*Since s* _{n} *> 2* ^{n} , the sum tends to ∞, and the series diverges. This is an example of unbounded divergence.

_{n}

^{n}

*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}

*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*

### =

### = ∑ = + + + ^{} +

### Main Menu

## 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.

### Main Menu

## 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*

### =

### = + =

### = + + =

### = + + + =

### = + + + + =

### Main Menu

## Infinite Series

### Main Menu

## Infinite Series

### Main Menu

### The Integral Test; Basic Comparison, Limit Comparison

### Main Menu

### The Integral Test; Basic Comparison, Limit Comparison

### Main Menu

### The Integral Test; Basic Comparison, Limit Comparison

### Main Menu

### 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

### Main Menu

### The Integral Test; Basic Comparison, Limit Comparison

### Main Menu

**Applying the Basic Comparison Theorem** **Example**

**(a)** converges by comparison with

**(b)** converges by comparison with

### 3

### 1 2 *k* + 1

### ∑ ∑ _{k} ^{1} ^{3}

_{k}

### 3 3 3

### 1 1 1

### and converges

### 2 *k* 1 < *k* *k*

### + ∑

### 3

### 5 4

### 5 7

*k* *k* + *k* +

### ∑ ∑ _{k} ^{1} ^{2}

_{k}

### 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

### Main Menu

### The Integral Test; Basic Comparison, Limit Comparison

### Main Menu

**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}

*k*

### ∞

### = −

### ∑

### 1 5 ^{k} − 3

^{k}

### 1 5 ^{k}

^{k}

### 1 5 ^{k}

^{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

### Main Menu

## The Root Test; The Ratio Test

### Main Menu

## The Root Test; The Ratio Test

**Example** For the series

### The series converges.

**Applying the Root Test**

### ( )

### 1 ln *k* ^{k}

^{k}

### ∑

### ( ) ^{1/} ^{1} ^{0}

### ln

*k*

*a* *k*

### = *k* →

### Main Menu

## The Root Test; The Ratio Test

### Main Menu

## The Root Test; The Ratio Test

**Applying the Ratio Test** **Example**

### For the series

### 10 ^{k}

^{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.

### Main Menu

## 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

^{p}

*integral test. Finally, keep in mind that, if a* _{k} 0, then there is no reason to try any convergence test; the series diverges.

_{k}

### → /

### Main Menu

## Absolute and Conditional Convergence

**Absolute Convergence and Conditional Convergence**

### Series *Σa* _{k} for which *Σ|a* _{k} *| converges are called absolutely convergent. The*

_{k}

_{k}

### theorem we have just proved says that

### Main Menu

## 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.

_{k}

### 1 1 1 1 1

### 1 − + − + − + 2 3 4 5 6

### ( )

### 0 1 2 3

### 0

### 1 ^{k} _{k}

^{k}

_{k}

*k*

*a* *a* *a* *a* *a*

### ∞

### =

### − + − + ^{} = ∑ −

### Main Menu

## 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}

_{k}

*k*

*a*

### ∞

### =

### ∑ −

### Main Menu

*Taylor Polynomials in x; Taylor Series in x*

### Main Menu

*Taylor Polynomials in x; Taylor Series in x*

*The following estimate for R* _{n} *(x) is an immediate consequence of * Corollary 12.6.2:

_{n}

*where J is the closed interval that joins 0 to x.*

### Main Menu

*Taylor Polynomials in x; Taylor Series 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*

### = ∑

### Main Menu

*Taylor Polynomials in x; Taylor Series in x*

### Main Menu

*Taylor Polynomials and Taylor Series in x – a*

### Main Menu

*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*

_{n}

*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*

### ′ ′′

### = + − + − + + −

### Main Menu

*Taylor Polynomials and Taylor Series in x – a*

*If R* _{n} *(x* ) → 0, then we have the series representation

_{n}

### 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.*

### Main Menu

## Power Series

### Main Menu

## 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**

**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.

^{k}

^{k}

**Case 2. The series converges absolutely at all real numbers x. This is what happens ** with the exponential series

**Case 2. The series converges absolutely at all real numbers x. This is what happens**

**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*

**Case 3. There exists a positive number r such that the series converges absolutely for |x| < r**