Section 11.1 The Least Upper Bound Axiom

a. Least Upper Bound Axiom

b. Examples

c. Theorem 11.1.2

d. Example

e. Greatest Lower Bound

f. Theorem 11.1.4

Section 11.2 Sequences of Real Numbers

a. Definition

b. Laws of Formation

c. Types of Sequences

d. Range of a Sequence

e. Increasing, Decreasing, Monotonic Sequences

f. Example

Section 11.3 Limit of A Sequence

a. Definition

b. Example

c. Uniqueness of Limit Theorem

d. Convergent and Divergent Sequences

e. Convergence Theorem

f. Example

g. Theorem 11.3.7

h. Pinching Theorem for Sequences

i. Corollary

j. Continuous Functions Applied to Convergent Sequences

### Chapter 11: Sequences; Indeterminate Forms; Improper Integrals

Section 11.4 Some Important Limits

a. Common Limits

b. Limit Properties

c. More Properties

Section 11.5 The Indeterminate Form (0/0)

a. L’Hôpitals Rule (0/0)

b. Example

c. The Cauchy Mean-Value Theorem

Section 11.6 The Indeterminate Form (∞/∞); Other Indeterminate Forms

a. L’Hôpitals Rule (∞/∞)

b. Limit Properties

c. Example

d. Indeterminates 0^{0}, 1^{∞}, ∞^{0}

e. Example

Section 11.7 Improper Integrals

a. Integrals Over Unbounded Intervals

b. Examples

c. Limit Property

d. Comparison Test

e. Integrals of Unbounded Functions

f. Example

### The Least Upper Bound Axiom

### The Least Upper Bound Axiom

*We indicate the least upper bound of a set S by writing lub S. As you will see *
from the examples below, the least upper bound idea has wide applicability.

**(1) ***lub (−∞, 0) = 0, lub(−∞, 0] = 0.*

**(2) ***lub (−4,−1) = −1, lub(−4,−1] = −1.*

**(3) lub {1/2, 2/3, 3/4, . . . , n/(n + 1), . . . } = 1.**

**(4) ***lub {−1/2,−1/8,−1/27, . . . ,−1/n*^{3}*, . . . } = 0.*

**(5) lub {x : x**^{2} *< 3} = lub{x : } = *
**(6) For each decimal fraction**

*b = 0.b*_{1}*b*_{2}*b*_{3}*, . . . ,*
we have

*b = lub {0.b*_{1}*, 0.b*_{1}*b*_{2}*, 0.b*_{1}*b*_{2}*b*_{3}*, . . . }.*

**(7) If S consists of the lengths of all polygonal paths inscribed in a semicircle ***of radius 1, then lub S = π (half the circumference of the unit circle).*

3 *x* 3

− < < 3

### The Least Upper Bound Axiom

### The Least Upper Bound Axiom

**Example**

**(a) Let S = {1/2, 2/3, 3/4, . . . , n/(n + 1), . . . } and take **ε = 0.0001. Since 1 *is the least upper bound of S, there must be a number s ∈ S such that*

*1 − 0.0001 < s < 1.*

There is: take, for example,

**(b) Let S = {0, 1, 2, 3} and take **ε = 0.00001. It is clear that 3 is the least *upper bound of S. Therefore, there must be a number s ∈ S such that*

*3 − 0.00001 < s ≤ 3.*

*There is: s = 3.*

99,999
100, 000.
*s* =

### The Least Upper Bound Axiom

### The Least Upper Bound Axiom

### Sequences of Real Numbers

### Sequences of Real Numbers

Suppose we have a sequence of real numbers

*a*_{1}*, a*_{2}*, a*_{3}*, . . . , a*_{n}*, . . . *

*By convention, a*_{1} *is called the first term of the sequence, a*_{2} *the second term, and *
*so on. More generally, a*_{n}*, the term with index n, is called the nth term.*

Sequences can be defined by giving the law of formation. For example:

2

1 1 1 2 3 4 1 2 3 4 2 3 4 5

1 is the sequence 1, , , , . . .

is the sequence , , , , . . . 1

is the sequence 1, 4, 9, 16, . . .

*n*

*n*

*n*

*a* *n*

*b* *n*

*n*
*c* *n*

=

= +

=

*It’s like defining f by giving f (x).*

### Sequences of Real Numbers

Sequences can be multiplied by constants; they can be added, subtracted, and multiplied. From

*a*_{1}*, a*_{2}*, a*_{3}*, . . . , a*_{n}*, . . . and b*_{1}*, b*_{2}*, b*_{3}*, . . . , b*_{n}*, . . .*
we can form

the scalar product sequence : *αa*_{1}*, αa*_{2}*, αa*_{3}*, . . . , αa*_{n}*, . . . ,*

*the sum sequence : a*_{1} *+ b*_{1}*, a*_{2} *+ b*_{2}*, a*_{3} *+ b*_{3}*, . . . , a*_{n}*+ b*_{n}*, . . . ,*
*the difference sequence : a*_{1} *− b*_{1}*, a*_{2} *− b*_{2}*, a*_{3} *− b*_{3}*, . . . , a*_{n}*− b*_{n}*, . . . ,*

*the product sequence : a*_{1}*b*_{1}*, a*_{2}*b*_{2}*, a*_{3}*b*_{3}*, . . . , a*_{n}*b*_{n}*, . . . .*

*If the b*_{i }*’s are all different from zero, we can form *

the quotient sequence : the reciprocal sequence :

1 2 3

1 1 1 1

, , , . . . , , . . .
*b b* *b* *b**n*

3 1 2

1 2 3

, , , . . . , * ^{n}* , . . .

*n*

*a* *a*

*a a*

*b b* *b* *b*

### Sequences of Real Numbers

*The range of a sequence is the set of values taken on by the sequence. While *
there can be repetition in a sequence, there can be no repetition in the

statement of its range. The range is a set, and in a set there is no repetition. A number is either in a particular set or it’s not. It can’t be there more than

once. The sequences

*0, 1, 0, 1, 0, 1, 0, 1, . . . and 0, 0, 1, 1, 0, 0, 1, 1, . . .*
*both have the same range: the set {0, 1}. The range of the sequence*

*0, 1,−1, 2, 2,−2, 3, 3, 3,−3, 4, 4, 4, 4,−4, . . .*
is the set of integers.

### Sequences of Real Numbers

Many of the sequences we work with have some regularity. They either have an upward tendency or they have a downward tendency. The following

*terminology is standard. The sequence with terms a** _{n}* is said to be

*increasing if a*

_{n}*< a*

_{n+1}*for all n,*

*nondecreasing if a*_{n}*≤ a*_{n+1}*for all n,*
*decreasing if a*_{n}*> a*_{n+1}*for all n,*
*nonincreasing if a*_{n}*≥ a*_{n+1}*for all n.*

*A sequence that satisfies any of these conditions is called monotonic.*

The sequences

*1, ½, *^{1}∕^{3}*, ¼, . . . , *^{1}∕^{n} *, . . .*
*2, 4, 8, 16, . . . , 2*^{n}*, . . .*
*2, 2, 4, 4, 6, 6, . . . , 2n, 2n, . . .*
are monotonic. The sequence

*1, ½, 1, *^{1}∕^{3}*, 1, ¼, 1, . . .*
is not monotonic.

### Sequences of Real Numbers

**Example**
The sequence

is increasing. It is bounded below by ½ (the greatest lower bound) and above by 1 (the least upper bound).

**Proof**
Since

*we have a*_{n}*< a** _{n+1}*. This confirms that the sequence is increasing. The sequence can be
displayed as

It is clear that ½ is the greatest lower bound and 1 is the least upper bound.

*n* 1
*a* *n*

= *n*
+

### ( ) ( )

### ( )

2 1

2

1 / 2 1 1 2 1

/ 1 2 2 1

*n*
*n*

*n* *n*

*a* *n* *n* *n* *n*

*a* *n* *n* *n* *n* *n* *n*

+ = + + = + ⋅ + = + + >

+ + +

1 2 3 4 98 99

, , , ,. . . , , , . . .

2 3 4 5 99 100

### Limit of A Sequence

### Limit of A Sequence

**Example**
Since

it is intuitively clear that

To verify that this statement conforms to the definition of limit of a sequence, we
must show that for each *ε > 0 there exists a positive integer K such that*

*if n ≥ K, then*

To do this, we fix *ε > 0 and note that*

*We now choose K sufficiently large that * *. If n ≥ K, then*
and consequently

2 2

1 1 1
*n*

*n* = *n*

+ +

lim 2 2

1

*n*

*n*

→∞ *n* =

+

2 2

1
*n*

*n* − <ε
+

### ( )

2 2 1

2

1 1

### 2 2 2

### 2

### 1 1

*n* *n*

*n*

*n* *n*

*n* *n* *n*

− +

+ +

### − = = − = <

### + +

2 2

1 2
*n*

*n* − < *n* < ε
+

2 *K* < ε 2 *n* ≤ 2 *K* <ε

### Limit of A Sequence

### Limit of A Sequence

### Limit of A Sequence

### Limit of A Sequence

**Example**

We shall show that the sequence is convergent. Since
3 = (3* ^{n}*)

^{1/n}*< (3*

*+ 4*

^{n}*)*

^{n}

^{1/n}*< (4*

*+ 4*

^{n}*)*

^{n}*= (2 · 4*

^{1/n}*)*

^{n}*= 2*

^{1/n}

^{1/n}*· 4 ≤ 8,*the sequence is bounded. Note that

(3* ^{n}* + 4

*= (3*

^{n)(n+1)/n}*+ 4*

^{n}*)*

^{n}*(3*

^{1/n}*+ 4*

^{n}*)*

^{n}= (3* ^{n}* + 4

*)*

^{n}*3*

^{1/n}*+ (3*

^{n}*+ 4*

^{n}*)*

^{n}*4*

^{1/n}

^{n}*.*Since

(3* ^{n}* + 4

*)*

^{n}

^{1/n}*> (3*

*)*

^{n}*= 3 and (3*

^{1/n}*+ 4*

^{n}*)*

^{n}

^{1/n}*> (4*

*)*

^{n}

^{1/n}*= 4,*we have

(3* ^{n}* + 4

*)*

^{n}

^{(n+1)/n}*> 3 · (3*

*) + 4 · (4*

^{n}*) = 3*

^{n}*+ 4*

^{n+1}

^{n+1}*.*

*Taking the (n + 1)st root of the left and right sides of this inequality, we obtain*
(3* ^{n}* + 4

*)*

^{n}

^{1/n}*> (3*

*+ 4*

^{n+1}*)*

^{n+1}

^{1/(n+1)}The sequence is decreasing. Being also bounded, it must be convergent.

### (

^{3}

^{n}^{4}

^{n}### )

^{1/}

^{n}*a**n* = +

### Limit of A Sequence

### Limit of A Sequence

### Limit of A Sequence

The following is an obvious corollary to the pinching theorem.

A limit to remember

### Limit of A Sequence

**Continuous Functions Applied to Convergent Sequences**

### Some Important Limits

### Some Important Limits

### Some Important Limits

### The Indeterminate Form (0/0)

### The Indeterminate Form (0/0)

**Example**
Find

**Solution**

*As x *→ 0^{+}, both numerator and denominator tend to 0 and

It follows from L’Hôpital’s rule that

For short, we can write

0

lim sin

*x*

*x*

+ *x*

→

### ( ) ( ) (

^{cos}

### ) (

^{1}

^{1/ 2}

## )

^{cos}

^{2}

^{1}

^{0}

^{0}

*f* *x* *x*

*g x* *x* *x* *x*

′ = = → =

′

0

lim 0

sin

*x*

*x*

+ *x*

→ →

0 0

lim lim 2 0

sin cos

*x* *x*

*x* *x*

*x* *x*

+ +

→ → =

### The Indeterminate Form (0/0)

### Other Indeterminate Forms

### Other Indeterminate Forms

### Other Indeterminate Forms

**Example**

*Determine the behavior of as n*→∞.

**Solution**

To use the methods of calculus, we investigate

*Since both numerator and denominator tend to ∞ with x, we try L’Hôpital’s rule:*

Therefore the sequence must also diverge to ∞.

2

2^{n}*a**n*

= *n*

2

lim 2

*x*

*x*^{→∞} *x*

### ( )

^{2}

2

2 ln 2

2 2 ln 2

lim lim lim

2 2

*x* *x* *x*

*x*^{→∞} *x* *x*^{→∞} *x* *x*^{→∞} = ∞

### Other Indeterminate Forms

**The Indeterminates 0**^{0}**, 1**^{∞}**, ∞**^{0}

Such indeterminates are usually handled by first applying the logarithm function:

*y = [ f (x)]*^{g(x)}*gives ln y = g(x) ln f (x).*

### Other Indeterminate Forms

**Example (1**^{∞})
Find

**Solution**

Here we are dealing with an indeterminate of the form 1^{∞}*: as x *→ 0^{+}*, 1 + x *→ 1
*and 1/x *→∞. Taking the logarithm and then applying L’Hôpital’s rule, we have

*As x *→ 0^{+}*, ln (1 + x)*^{1/x}*→ 1 and (1 + x)*^{1/x}*= e*^{ln(1+x)}^{1/x}*→ e*^{1} *= e. Set x = 1/n*
*and we have the familiar result: as n→∞, [1 + (1/n)]*^{n}*→ e.*

### ( )

^{1/}

0

### lim 1

^{x}*x*

+

*x*

→

### +

### ( )

^{1/}

### ( )

0 0 0

### ln 1 1

### lim ln 1 lim lim 1

### 1

*x*

*x* *x* *x*

*x* *x*

*x* *x*

+ + +

→ → →

### + = + =

### +

### Improper Integrals

**Integrals over Unbounded Intervals**

*We begin with a function f which is continuous on an unbounded interval [a,*∞). For
*each number b > a we can form the definite integral*

*If, as b tends to ∞, this integral tends to a finite limit L,*

### ( )

*b*

*a* *f x dx*

### ∫

### ( )

lim ^{b}

*b* *a* *f x dx* *L*

→∞

### ∫

=then we write

and say that

*the improper integral*

Otherwise, we say that

*the improper integral*

*converges to L.*

*diverges.*

*a*^{∞} *f x dx*

### ( )

=*L*

### ∫

*a*^{∞} *f x dx*

### ( )

### ∫

*a*^{∞} *f x dx*

### ( )

### ∫

### Improper Integrals

*As a tends to −∞, sin πa oscillates between −1 and 1. Therefore the integral *
oscillates between 1*/π and −1/π and does not converge.*

### Improper Integrals

### Improper Integrals

### Improper Integrals

**Integrals of Unbounded Functions**

*Improper integrals can arise on bounded intervals. Suppose that f *
*is continuous on the half-open interval [a, b) but is unbounded *
*there. For each number c < b, we can form the definite integral*

*If, as c* *→ b*^{−}*, the integral tends to a finite limit L, namely, if*

### ( )

*c*

*a* *f x dx*

### ∫

### ( )

lim ^{c}

*c* *b* *a*

*f x dx* *L*

→ −

### ∫

=then we write

and say that

*the improper integral*

*Otherwise, we say that the improper integral diverges.*

*converges to L.*

### ( )

*b*

*a* *f x dx* = *L*

### ∫

### ( )

*b*

*a* *f x dx*

### ∫

### Improper Integrals

**Example**
Test

(∗)

for convergence.

**Solution**

*The integrand has an infinite discontinuity at x = 2.*

For integral (∗) to converge both

*must converge. Neither does. For instance, as c *→ 2^{−},

This tells us that

*If we overlook the infinite discontinuity at x = 2, we can be led to the incorrect *
*conclusion that*

### ( )

4 1 2

2
*dx*
*x* −

### ∫

### ( ) ( )

2 4

2 2

1 and 2

2 2

*dx* *dx*

*x*− *x*−

### ∫ ∫

### ( )

^{2}

1 1

1 1

2 2 1

2

*c* *dx* *c*

*x* *c*

*x*

= − − = − − − → ∞

### ∫

−### ( )

2 1 2

2
*dx*
*x* −

### ∫

diverges and shows that (∗) diverges.### ( )

4 4

1 2

1

1 3

2 2

2
*dx*
*x* *x*

= − − = −