Monotone Sequences Objective: To define a Monotone Sequence and determine whether it converges or not.

Monotone Sequences

Objective: To define a Monotone Sequence and determine whether it

converges or not.

Monotone Sequences

• We will begin with some terminology.• Definition 9.2.1 A sequence is called

• Strictly increasing if• Increasing if• Strictly decreasing if• Decreasing if

1nna

......321 naaaa

......321 naaaa

......321 naaaa

......321 naaaa

Monotone Sequences

• We will begin with some terminology.• Definition 9.2.1 A sequence is called

• Strictly increasing if• Increasing if• Strictly decreasing if• Decreasing if• A sequence that is either increasing or decreasing is

said to be monotone, and a sequence that is either strictly increasing or strictly decreasing is said to be strictly monotone.

......321 naaaa

......321 naaaa

......321 naaaa

......321 naaaa

1nna

Monotone Sequences

• Some examples are:

Monotone Sequences

• Some examples are:

Testing for Monotonicity

• Frequently, one can guess whether a sequence is monotone or strictly monotone by writing out some of the initial terms. However, to be certain that the guess is correct, one must give a precise mathematical argument. We will look at two different ways to accomplish this.

Testing for Monotonicity

• Frequently, one can guess whether a sequence is monotone or strictly monotone by writing out some of the initial terms. However, to be certain that the guess is correct, one must give a precise mathematical argument. We will look at two different ways to accomplish this.

Example 1

• Use both methods to show that the following is a strictly increasing sequence.

,...1

,...,4

3,3

2,2

1

nn

Example 1

• Use both methods to show that the following is a strictly increasing sequence.

,...1

,...,4

3,3

2,2

1

nn

1n

nan

2

1

1)1(

11

n

n

n

nan

Example 1

• Use both methods to show that the following is a strictly increasing sequence. (Difference of terms)

,...1

,...,4

3,3

2,2

1

nn

1n

nan

2

1

1)1(

11

n

n

n

nan

012

1

n

n

n

n

Example 1

• Use both methods to show that the following is a strictly increasing sequence. (Difference of terms)

,...1

,...,4

3,3

2,2

1

nn

1n

nan

2

1

1)1(

11

n

n

n

nan

012

1

n

n

n

n

0)2)(1(

1

2

2

11

1

2

1

nnn

n

n

n

n

n

n

n

Example 2

• Use both methods to show that the following is a strictly increasing sequence. (Ratio of terms)

,...1

,...,4

3,3

2,2

1

nn

1n

nan

2

1

1)1(

11

n

n

n

nan

12

121

2

1

)1/(

)2/()1(2

21

nn

nn

n

n

n

n

nn

nn

a

a

n

n

Example 3

• There is a third method we can use. We need to look at the sequence as a function and apply the first derivative test.

1)(

x

xxf

Example 3

• There is a third method we can use. We need to look at the sequence as a function and apply the first derivative test.

• If the first derivative is positive, the function is increasing everywhere, thus strictly increasing.

1)(

x

xxf

0)1(

1

)1(

1)(

22/

xx

xxxf

Properties that hold Eventually

• Sometimes a sequence will behave erratically at first and then settle down into a definite pattern. For example, the sequence 9, -8, -17, 12, 1, 2, 3, 4,… is strictly increasing from the fifth term on, but the sequence as a whole cannot be classified as strictly increasing because of the erratic behavior of the first four terms. To describe such sequences, we will introduce the following terminology.

Properties that hold Eventually

• Definition 9.2.2 • If discarding finitely many terms from the beginning

of a sequence produces a sequence with a certain property, then the original sequence is said to have that property eventually.

Example 4

• Show that the sequence is eventually strictly decreasing.

1!

10

n

n

n

Example 4

• Show that the sequence is eventually strictly decreasing.

1!

10

n

n

n

!

10

na

n

n

)!1(

10 1

1

na

n

n

Example 4

• Show that the sequence is eventually strictly decreasing.

1!

10

n

n

n

!

10

na

n

n

)!1(

10 1

1

na

n

n

n

n

n

n n

nn

n

n 10

!

!)1(

1010

10

!

)!1(

10 1

Example 4

• Show that the sequence is eventually strictly decreasing.

• To be decreasing, this ratio needs to be less than 1. This will occur for n > 10, so the sequence is eventually strictly decreasing.

1!

10

n

n

n

1

10

10

!

!)1(

1010

10

!

)!1(

10 1

n

n

nn

n

n n

n

n

n

Intuitive View of Convergence

• Informally stated, the convergence or divergence of a sequence does not depend on the behavior of its initial terms, but rather on how the terms behave eventually. For example, the sequence

• eventually behaves like the sequence

• and hence has a limit of 0.

,...4

1,3

1,2

1,1,17,13,9,3

,...1

,...,3

1,2

1,1

n

Convergence of Monotone Sequences

• The following theorems show that a monotone sequence either converges or becomes infinite-divergence by oscillation cannot occur.

Convergence of Monotone Sequences

• The following theorems show that a monotone sequence either converges or becomes infinite-divergence by oscillation cannot occur.

Example 5

• Show that the sequence converges and find its limit.

1!

10

n

n

n

Example 5

• Show that the sequence converges and find its limit.

• We proved that this is eventually strictly decreasing. Since all terms are positive, it is bounded below by M = 0, and hence the theorem guarantees that it converges to a nonnegative limit L. We will look at it this way:

1!

10

n

n

n

Example 5

• Show that the sequence converges and find its limit.

• We proved that this is eventually strictly decreasing. Since all terms are positive, it is bounded below by M = 0, and hence the theorem guarantees that it converges to a nonnegative limit L. We will look at it this way:

1!

10

n

n

n

nn an

a1

101 Laa n

nn

n

limlim 1

1

101

na

a

n

n

Example 5

• Show that the sequence converges and find its limit.

• We proved that this is eventually strictly decreasing. Since all terms are positive, it is bounded below by M = 0, and hence the theorem guarantees that it converges to a nonnegative limit L. We will look at it this way:

1!

10

n

n

n

nn an

a1

101 Laa n

nn

n

limlim 1

00lim1

10lim

1

10lim

La

na

n nnn

nn

Homework

• Section 9.2• Page 613• 1-21 odd