23 monotone sequences

Monotone Sequences

Let {an} be a sequence, we say it is* increasing if an< an+1

Monotone Sequences

Let {an} be a sequence, we say it is* increasing if an< an+1

* nondecreasing if an < an+1

Monotone Sequences

Let {an} be a sequence, we say it is* increasing if an< an+1

* nondecreasing if an < an+1

* decreasing if an > an+1

Monotone Sequences

Let {an} be a sequence, we say it is* increasing if an< an+1

* nondecreasing if an < an+1

* decreasing if an > an+1 * nonincreasing an > an+1

Monotone Sequences

Let {an} be a sequence, we say it is* increasing if an< an+1

* nondecreasing if an < an+1

* decreasing if an > an+1 * nonincreasing an > an+1

Example: 1, 2, 3, 4…. is an increasing sequence

Monotone Sequences

Let {an} be a sequence, we say it is* increasing if an< an+1

* nondecreasing if an < an+1

* decreasing if an > an+1 * nonincreasing an > an+1

Example: 1, 2, 3, 4…. is an increasing sequence1, 2, 3, 3, 3, ….is a nondecreasing sequence

Monotone Sequences

Let {an} be a sequence, we say it is* increasing if an< an+1

* nondecreasing if an < an+1

* decreasing if an > an+1 * nonincreasing an > an+1

Example: 1, 2, 3, 4…. is an increasing sequence1, 2, 3, 3, 3, ….is a nondecreasing sequence-1, -2, -3, -4, …is decreasing sequence-1, -2, -3, -3, -3, ….is a nonincreasing sequence

Monotone Sequences

Let {an} be a sequence, we say it is* increasing if an< an+1

* nondecreasing if an < an+1

* decreasing if an > an+1 * nonincreasing an > an+1

Example: 1, 2, 3, 4…. is an increasing sequence1, 2, 3, 3, 3, ….is a nondecreasing sequence-1, -2, -3, -4, …is decreasing sequence-1, -2, -3, -3, -3, ….is a nonincreasing sequence1, -2, 3, -4… is none of these.

Monotone Sequences

Let {an} be a sequence, we say it is* increasing if an< an+1

* nondecreasing if an < an+1

* decreasing if an > an+1 * nonincreasing an > an+1

Example: 1, 2, 3, 4…. is an increasing sequence1, 2, 3, 3, 3, ….is a nondecreasing sequence-1, -2, -3, -4, …is decreasing sequence-1, -2, -3, -3, -3, ….is a nonincreasing sequence1, -2, 3, -4… is none of these.

We call these four types of sequences monotonesequences.

Monotone Sequences

Let {an} be a sequence, we say it is* increasing if an< an+1

* nondecreasing if an < an+1

* decreasing if an > an+1 * nonincreasing an > an+1

Example: 1, 2, 3, 4…. is an increasing sequence1, 2, 3, 3, 3, ….is a nondecreasing sequence-1, -2, -3, -4, …is decreasing sequence-1, -2, -3, -3, -3, ….is a nonincreasing sequence1, -2, 3, -4… is none of these.

We call these four types of sequences monotonesequences. Increasing and decreasing sequences are said to be strictly monotone.

Monotone Sequences

Although sometime its easy to identify if a given sequence is monotone, it is important to find formal ways to justify the claim mathematically.

Monotone Sequences

Although sometime its easy to identify if a given sequence is monotone, it is important to find formal ways to justify the claim mathematically. Following are three basic methods.

Monotone Sequences

Theorem: {f(n)} = {an}n=1 is increasing if and only if I. an+1 – an > 0 for all n,

Although sometime its easy to identify if a given sequence is monotone, it is important to find formal ways to justify the claim mathematically. Following are three basic methods.

Monotone Sequences

∞

Theorem: {f(n)} = {an}n=1 is increasing if and only if I. an+1 – an > 0 for all n, orII. an+1 / an > 1 for all n,

Although sometime its easy to identify if a given sequence is monotone, it is important to find formal ways to justify the claim mathematically. Following are three basic methods.

Monotone Sequences

∞

Theorem: {f(n)} = {an}n=1 is increasing if and only if I. an+1 – an > 0 for all n, orII. an+1 / an > 1 for all n, orIII. f '(x) > 0 for x > 0

Although sometime its easy to identify if a given sequence is monotone, it is important to find formal ways to justify the claim mathematically. Following are three basic methods.

Monotone Sequences

∞

Theorem: {f(n)} = {an}n=1 is increasing if and only if I. an+1 – an > 0 for all n, orII. an+1 / an > 1 for all n, orIII. f '(x) > 0 for x > 0

Although sometime its easy to identify if a given sequence is monotone, it is important to find formal ways to justify the claim mathematically. Following are three basic methods.

Monotone Sequences

∞

In the above theorem, we may replace the inequality with < for decreasing sequences.

Theorem: {f(n)} = {an}n=1 is increasing if and only if I. an+1 – an > 0 for all n, orII. an+1 / an > 1 for all n, orIII. f '(x) > 0 for x > 0

Although sometime its easy to identify if a given sequence is monotone, it is important to find formal ways to justify the claim mathematically. Following are three basic methods.

Monotone Sequences

∞

In the above theorem, we may replace the inequality with < for decreasing sequences. Replace it with > for nondecreasing sequences.

Theorem: {f(n)} = {an}n=1 is increasing if and only if I. an+1 – an > 0 for all n, orII. an+1 / an > 1 for all n, orIII. f '(x) > 0 for x > 0

Although sometime its easy to identify if a given sequence is monotone, it is important to find formal ways to justify the claim mathematically. Following are three basic methods.

Monotone Sequences

∞

In the above theorem, we may replace the inequality with < for decreasing sequences. Replace it with > for nondecreasing sequences.Replace it with < for nonincreasing sequences.

Example: Justify that {an} = {2n/n!} is a decreasing sequences for n = 2, 3, 4, ...

Monotone Sequences

Example: Justify that {an} = {2n/n!} is a decreasing sequences for n = 2, 3, 4, ...The most suitable method is the quotient method.

Monotone Sequences

Example: Justify that {an} = {2n/n!} is a decreasing sequences for n = 2, 3, 4, ...The most suitable method is the quotient method.

an+1 =

Monotone Sequences

n!2n(n+1)!

2n+1* =

an

1

Example: Justify that {an} = {2n/n!} is a decreasing sequences for n = 2, 3, 4, ...The most suitable method is the quotient method.

an+1 =

Monotone Sequences

n!2n(n+1)!

2n+1* = 2

n+1an

1

Example: Justify that {an} = {2n/n!} is a decreasing sequences for n = 2, 3, 4, ...The most suitable method is the quotient method.

an+1 =

Monotone Sequences

n!2n(n+1)!

2n+1* = 2

n+1< 1 for n = 2, 3, …

an

1

Example: Justify that {an} = {2n/n!} is a decreasing sequences for n = 2, 3, 4, ...The most suitable method is the quotient method.

an+1 =

Monotone Sequences

n!2n(n+1)!

2n+1* = 2

n+1< 1 for n = 2, 3, …

an

1

So {2n/n!} is a decreasing sequence where n > 1.

Example: Justify that {an} = {2n/n!} is a decreasing sequences for n = 2, 3, 4, ...The most suitable method is the quotient method.

The behavior of a sequence is determined by the "tail" of the sequence.

an+1 =

Monotone Sequences

n!2n(n+1)!

2n+1* = 2

n+1< 1 for n = 2, 3, …

an

1

So {2n/n!} is a decreasing sequence where n > 1.

Example: Justify that {an} = {2n/n!} is a decreasing sequences for n = 2, 3, 4, ...The most suitable method is the quotient method.

The behavior of a sequence is determined by the "tail" of the sequence. We say a sequence is eventually monotone if the sequence is monotone for all an’s where n > k for some k.

an+1 =

Monotone Sequences

n!2n(n+1)!

2n+1* = 2

n+1< 1 for n = 2, 3, …

an

1

So {2n/n!} is a decreasing sequence where n > 1.

Example: Justify that {an} = {2n/n!} is a decreasing sequences for n = 2, 3, 4, ...The most suitable method is the quotient method.

The behavior of a sequence is determined by the "tail" of the sequence. We say a sequence is eventually monotone if the sequence is monotone for all an’s where n > k for some k. The sequence 1, 0, 2, 0, 3, 4, 5, 6, 7… is eventually increasing because it is increasing beyond the 5’th term (k = 5).

an+1 =

Monotone Sequences

n!2n(n+1)!

2n+1* = 2

n+1< 1 for n = 2, 3, …

an

1

So {2n/n!} is a decreasing sequence where n > 1.

Example: Justify that {an} = {2n/n!} is a decreasing sequences for n = 2, 3, 4, ...The most suitable method is the quotient method.

The behavior of a sequence is determined by the "tail" of the sequence. We say a sequence is eventually monotone if the sequence is monotone for all an’s where n > k for some k. The sequence 1, 0, 2, 0, 3, 4, 5, 6, 7… is eventually increasing because it is increasing beyond the 5’th term (k = 5). We may also say that the sequence is increasing “for large n”,

an+1 =

Monotone Sequences

n!2n(n+1)!

2n+1* = 2

n+1< 1 for n = 2, 3, …

an

1

So {2n/n!} is a decreasing sequence where n > 1.

Example: Justify that {an} = {2n/n!} is a decreasing sequences for n = 2, 3, 4, ...The most suitable method is the quotient method.

The behavior of a sequence is determined by the "tail" of the sequence. We say a sequence is eventually monotone if the sequence is monotone for all an’s where n > k for some k. The sequence 1, 0, 2, 0, 3, 4, 5, 6, 7… is eventually increasing because it is increasing beyond the 5’th term (k = 5). We may also say that the sequence is increasing “for large n”, i.e. for all an’s where n > k for some k.

an+1 =

Monotone Sequences

n!2n(n+1)!

2n+1* = 2

n+1< 1 for n = 2, 3, …

an

1

So {2n/n!} is a decreasing sequence where n > 1.

A sequence {an} is said to be bounded above if there exists a number A such that A > an for all an.

Monotone Sequences

A sequence {an} is said to be bounded above if there exists a number A such that A > an for all an.

Monotone Sequences

A sequence {an} is said to be bounded below if there exists a number B such that B < an for all an.

A sequence {an} is said to be bounded above if there exists a number A such that A > an for all an.

Monotone Sequences

A sequence {an} is said to be bounded below if there exists a number B such that B < an for all an.

A sequence {an} is said to be bounded if it is both bounded above and below.

A sequence {an} is said to be bounded above if there exists a number A such that A > an for all an.

Theorem: Let {an} be an eventually monotone sequence.

Monotone Sequences

A sequence {an} is said to be bounded below if there exists a number B such that B < an for all an.

A sequence {an} is said to be bounded if it is both bounded above and below.

A sequence {an} is said to be bounded above if there exists a number A such that A > an for all an.

Theorem: Let {an} be an eventually monotone sequence. * If {an} is bounded, then it converges, i.e. lim an = Lfor some finite L.

Monotone Sequences

A sequence {an} is said to be bounded below if there exists a number B such that B < an for all an.

A sequence {an} is said to be bounded if it is both bounded above and below.

n∞

A sequence {an} is said to be bounded above if there exists a number A such that A > an for all an.

Theorem: Let {an} be an eventually monotone sequence. * If {an} is bounded, then it converges, i.e. lim an = Lfor some finite L.* If {an} is not bounded then lim an = + or – .

Monotone Sequences

A sequence {an} is said to be bounded below if there exists a number B such that B < an for all an.

A sequence {an} is said to be bounded if it is both bounded above and below.

n∞

n∞

∞ ∞

Hence,a nondecreasing sequence that is bounded above is a CG sequence.

Monotone Sequences

Hence,a nondecreasing sequence that is bounded above is a CG sequence.

Monotone Sequences

the upperbound A

a1 a2 a3 a4 . ..

Hence,a nondecreasing sequence that is bounded above is a CG sequence.

Monotone Sequences

the upperbound A

a1 a2 a3 a4 . ..

A nonincreasing sequence that is bounded below is a CG sequence.

Hence,a nondecreasing sequence that is bounded above is a CG sequence.

Monotone Sequences

the upperbound A

a1 a2 a3 a4 . ..

A nonincreasing sequence that is bounded below is a CG sequence.

the lowerbound b

a1a2a3... a4

Hence,a nondecreasing sequence that is bounded above is a CG sequence.

Monotone Sequences

the upperbound A

a1 a2 a3 a4 . ..

A nonincreasing sequence that is bounded below is a CG sequence.

the lowerbound b

a1a2a3... a4

Note the theorem just gives the existence of the limit but it doesn't indicate what the limit is.

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

We will show that an+1 – an < 0:

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

2n+1

1 + 3n+1– 2n

1 + 3n

We will show that an+1 – an < 0:

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

2n+1

1 + 3n+1– 2n

1 + 3nLCD = (1 + 3n+1)(1 + 3n)

We will show that an+1 – an < 0:

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

2n+1

1 + 3n+1– 2n

1 + 3n

=

LCD = (1 + 3n+1)(1 + 3n)

2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD

We will show that an+1 – an < 0:

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

2n+1

1 + 3n+1– 2n

1 + 3n

=

LCD = (1 + 3n+1)(1 + 3n)

2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD

= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD

We will show that an+1 – an < 0:

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

2n+1

1 + 3n+1– 2n

1 + 3n

=

LCD = (1 + 3n+1)(1 + 3n)

2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD

= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD

= 2n[2 + 2*3n – 1 – 3n+1] / LCD

We will show that an+1 – an < 0:

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

2n+1

1 + 3n+1– 2n

1 + 3n

=

LCD = (1 + 3n+1)(1 + 3n)

2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD

= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD

= 2n[2 + 2*3n – 1 – 3n+1] / LCD

= 2n[1 + 3n (2 – 3)] / LCD

We will show that an+1 – an < 0:

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

2n+1

1 + 3n+1– 2n

1 + 3n

=

LCD = (1 + 3n+1)(1 + 3n)

2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD

= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD

= 2n[2 + 2*3n – 1 – 3n+1] / LCD

= 2n[1 + 3n (2 – 3)] / LCD

= 2n[1 – 3n ] / LCD < 0 for n = 1, 2, 3 ..

We will show that an+1 – an < 0:

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

2n+1

1 + 3n+1– 2n

1 + 3n

=

LCD = (1 + 3n+1)(1 + 3n)

2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD

= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD

= 2n[2 + 2*3n – 1 – 3n+1] / LCD

= 2n[1 + 3n (2 – 3)] / LCD

= 2n[1 – 3n ] / LCD < 0 for n = 1, 2, 3 ..

We will show that an+1 – an < 0:

So the sequence is decreasing.

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

2n+1

1 + 3n+1– 2n

1 + 3n

=

LCD = (1 + 3n+1)(1 + 3n)

2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD

= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD

= 2n[2 + 2*3n – 1 – 3n+1] / LCD

= 2n[1 + 3n (2 – 3)] / LCD

= 2n[1 – 3n ] / LCD < 0 for n = 1, 2, 3 ..

We will show that an+1 – an < 0:

So the sequence is decreasing. All the terms are positive so its bounded below by 0.

Example: Show { } is a convergent sequence.2n

1 + 3n

Monotone Sequences

2n+1

1 + 3n+1– 2n

1 + 3n

=

LCD = (1 + 3n+1)(1 + 3n)

2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD

= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD

= 2n[2 + 2*3n – 1 – 3n+1] / LCD

= 2n[1 + 3n (2 – 3)] / LCD

= 2n[1 – 3n ] / LCD < 0 for n = 1, 2, 3 ..

We will show that an+1 – an < 0:

So the sequence is decreasing. All the terms are positive so its bounded below by 0. Hence it’s a CG sequence.