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Monotone Sequences
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Let {an} b= e a sequence, we say it is
* increasing if  an<= ; an+1
Monotone Sequences
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Let {an} b= e a sequence, we say it is
* increasing if  an<= ; an+1
* nondecreasing if an < an+1
Monotone Sequences
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Let {an} b= e a sequence, we say it is
* increasing if  an<= ; an+1
* nondecreasing if an < an+1
* decreasing if an > an+1 
Monotone Sequences
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Let {an} b= e 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
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Let {an} b= e 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
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Let {an} b= e 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
1, 2, 3, 3, 3, ….is a nondecreasing sequence
Monotone Sequences
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Let {an} b= e 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
1, 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
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Let {an} b= e 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
1, 2, 3, 3, 3, ….is a nondecreasing sequence
-1, -2, -3, -4, …is decreasing sequence
-1, -2, -3, -3, -3, ….is= a nonincreasing sequence
1, -2, 3, -4… is none of these.
Monotone Sequences
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Let {an} b= e 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
1, 2, 3, 3, 3, ….is a nondecreasing sequence
-1, -2, -3, -4, …is decreasing sequence
-1, -2, -3, -3, -3, ….is= a nonincreasing sequence
1, -2, 3, -4… is none of these.
We call these four types of sequences monotone
sequences. 
Monotone Sequences
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Let {an} b= e 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
1, 2, 3, 3, 3, ….is a nondecreasing sequence
-1, -2, -3, -4, …is decreasing sequence
-1, -2, -3, -3, -3, ….is= a nonincreasing sequence
1, -2, 3, -4… is none of these.
We call these four types of sequences monotone
sequences.  Increasing and decreasing sequenc= es are said to be strictly monotone= .
Monotone Sequences
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Alt= hough 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
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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
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Theorem: {f(n)} =3D {an}n=3D1 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
∞
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Theorem: {f(n)} =3D {an}n=3D1 is increasing if and only if  
I. an+1 –  an > 0 for all n,
        = ;        or
II. an+1 / a= n > 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
∞
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Theorem: {f(n)} =3D {an}n=3D1 is increasing if and only if  
I. an+1 –  an > 0 for all n,
        = ;        or
II. an+1 / a= n > 1 for all n,
        = ;        or
III.&nbs= p; 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
∞
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Theorem: {f(n)} =3D {an}n=3D1 is increasing if and only if  
I. an+1 –  an > 0 for all n,
        = ;        or
II. an+1 / a= n > 1 for all n,
        = ;        or
III.&nbs= p; 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. 
 
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Theorem: {f(n)} =3D {an}n=3D1 is increasing if and only if  
I. an+1 –  an > 0 for all n,
        = ;        or
II. an+1 / a= n > 1 for all n,
        = ;        or
III.&nbs= p; 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.
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Theorem: {f(n)} =3D {an}n=3D1 is increasing if and only if  
I. an+1 –  an > 0 for all n,
        = ;        or
II. an+1 / a= n > 1 for all n,
        =         or
III.&nbs= p; 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.
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Example: Justify that {an} =3D {2n/n!} is a decreasing sequences for n =3D 2, 3, 4, .= ..
Monotone Sequences
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Example: Justify that {an} =3D {2n/n!} is a decreasing sequences for n =3D 2, 3, 4, .= ..
Of the three method, the algeb= ra most suitable for
dealing with factorial is the quotient method because the cancellation of factorials= .
Monotone Sequences
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Example: Justify that {an} =3D {2n/n!} is a decreasing sequences for n =3D 2, 3, 4, .= ..
Of the three method, the algeb= ra most suitable for
dealing with factorial is the quotient method because the cancellation of factorials= .
an+1/an =3D
Monotone Sequences
(n+1)!
2n+1
*
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Example: Justify that {an} =3D {2n/n!} is a decreasing sequences for n =3D 2, 3, 4, .= ..
Of the three method, the algeb= ra most suitable for
dealing with factorial is the quotient method because the cancellation of factorials= .
an+1/an =3D
Monotone Sequences
n!
2n
(n+1)!
2n+1
*
 =3D
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Example: Justify that {an} =3D {2n/n!} is a decreasing sequences for n =3D 2, 3, 4, .= ..
Of the three method, the algeb= ra most suitable for
dealing with factorial is the quotient method because the cancellation of factorials= .
an+1/an =3D
Monotone Sequences
n!
2n
(n+1)!
2n+1
*
 =3D
2
n+1
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0120_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABm AAEAgAAAAAAAAAIMjI+py40Ao5wUuOsKADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0120_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAf AAEAgAAAAAAAAAIFjI+pewUAOw== ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0121.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Example: Justify that {an} =3D {2n/n!} is a decreasing sequences for n =3D 2, 3, 4, .= ..
Of the three method, the algeb= ra most suitable for
dealing with factorial is the quotient method because the cancellation of factorials= .
an+1/an =3D
Monotone Sequences
n!
2n
(n+1)!
2n+1
*
 =3D
2
n+1
< 1 for n =3D 2, = 3, …=
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0121_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABm AAEAgAAAAAAAAAIMjI+py40Ao5wUuOsKADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0121_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAf AAEAgAAAAAAAAAIFjI+pewUAOw== ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0122.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Example: Justify that {an} =3D {2n/n!} is a decreasing sequences for n =3D 2, 3, 4, .= ..
Of the three method, the algeb= ra most suitable for
dealing with factorial is the quotient method because the cancellation of factorials= .
The behavior of a sequence is determined by the "tail" of the sequenc= e.
an+1/an =3D
Monotone Sequences
n!
2n
(n+1)!
2n+1
*
 =3D
2
n+1
< 1 for n =3D 2, = 3, …=
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0122_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABm AAEAgAAAAAAAAAIMjI+py40Ao5wUuOsKADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0122_image027.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAf AAEAgAAAAAAAAAIFjI+pewUAOw== ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0123.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Example: Justify that {an} =3D {2n/n!} is a decreasing sequences for n =3D 2, 3, 4, .= ..
Of the three method, the algeb= ra most suitable for
dealing with factorial is the quotient method because the cancellation of factorials= .
The behavior of a sequence is determined by the "tail" of the sequenc= e. We say a sequence is eventually monotone if the sequence is monotone for all an where n > k for some k.<= span lang=3DEN-US style=3D'font-family:SimSun;mso-fareast-font-family:SimSun; mso-hansi-font-family:Arial;font-size:222%;mso-fareast-language:ZH-CN; mso-special-format:lastCR;display:none'>
an+1/an =3D
Monotone Sequences
n!
2n
(n+1)!
2n+1
*
 =3D
2
n+1
< 1 for n =3D 2, = 3, …=
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0123_image028.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABm AAEAgAAAAAAAAAIMjI+py40Ao5wUuOsKADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0123_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAf AAEAgAAAAAAAAAIFjI+pewUAOw== ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0124.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Example: Justify that {an} =3D {2n/n!} is a decreasing sequences for n =3D 2, 3, 4, .= ..
Of the three method, the algeb= ra most suitable for
dealing with factorial is the quotient method because the cancellation of factorials= .
The behavior of a sequence is determined by the "tail" of the sequenc= e. We say a sequence is eventually monotone if the sequence is monotone for all an where n > k for some k.<= span lang=3DEN-US style=3D'font-family:SimSun;mso-fareast-font-family:SimSun; mso-hansi-font-family:Arial;font-size:222%;mso-fareast-language:ZH-CN; mso-special-format:lastCR;display:none'>
an+1/an =3D
Example: The sequence 1, 0, 2,= 0, 3, 4, 5, 6, 7…
is a eventually increasing sequence= .
Monotone Sequences
n!
2n
(n+1)!
2n+1
*
 =3D
2
n+1
< 1 for n =3D 2, = 3, …=
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0124_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABm AAEAgAAAAAAAAAIMjI+py40Ao5wUuOsKADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0124_image031.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAf AAEAgAAAAAAAAAIFjI+pewUAOw== ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0098.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" 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
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0125.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence {an} is said to be bounded above if <= /span>there exists a number A such th= at 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 &= lt; an for all an.
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0125_image032.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0125_image033.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0126.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence {an} is said to be bounded above if <= /span>there exists a number A such th= at 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 &= lt; an for all an.
A sequence {an} is said to be bounded if it is both = bounded above and below.=
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0126_image034.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0126_image035.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0127.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence {an} is said to be bounded above if <= /span>there exists a number A such th= at A > an for all an.
The= orem: 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 &= lt; an for all an.
A sequence {an} is said to be bounded if it is both = bounded above and below.=
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0127_image036.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0127_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0128.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence {an} is said to be bounded above if <= /span>there exists a number A such th= at A > an for all an.
Theorem: Let {an} b= e an eventually monotone sequence.
* If {an} is bounded, then it converges, i.e.  lim an =3D L
for some finite L.
Monotone Sequences
A sequence {an} is said to be bounded below if there = exists a number B such that B &= lt; an for all an.
A sequence {an} is said to be bounded if it is both = bounded above and below.=
nà= ∞
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0128_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0128_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0129.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence {an} is said to be bounded above if <= /span>there exists a number A such th= at A > an for all an.
Theorem: Let {an} b= e an eventually monotone sequence.
* If {an} is bounded, then it converges, i.e.  lim an =3D L
for some finite L.
* If {an} is not bounded then lim an =3D +     or  –    .
Monotone Sequences
A sequence {an} is said to be bounded below if there = exists a number B such that B &= lt; an for all an.
A sequence {an} is said to be bounded if it is both = bounded above and below.=
nà= ∞
nà= ∞
∞
∞
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0129_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0129_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0099.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Hence,
a nondecreasing sequence that is bounded above is a CG sequence.
Monotone Sequences
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0130.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Hence,
a nondecreasing sequence that is bounded above is a CG sequence.
Monotone Sequences
the upper
bound A
a1<= /span>
a2<= /span>
a3<= /span>
a4<= /span>  . ..
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0130_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1gEDAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQDT AQIAgAAAAAAAAAIdjI+py+0Po5y02ouz3rz7D4biSJbmiabqyrbuWAAAOw== ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0130_image043.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAwAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAC ACQAgAAAAAAAAAIHjI+py+2vCgA7 ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0130_image044.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAALvg4wECAwIYhB2nEcIrYkygSZnsdXrnPVWgg3gL8wAFADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0130_image045.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhggAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQB9 AAoAgQAAAAAAALvg4/8AAAJ7hI+pyxgNzYtp0mOv3hwHEWRXCIYUaUbhl1LD0MXQJ9TgRtuikte7 Y9NdXi+ZERMUnpK+Rk+5eDYjRCKiis1qt9yu9wsOi7mGsfmMTqvX7Lb7bT52pLcl8ydh1qN6/BUr J/Pkx5NECDCoYjhkFWi00qIBeSgBOWLJyFAAADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0131.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Hence,
a nondecreasing sequence that is bounded above is a CG sequence.
Monotone Sequences
the upper
bound A
a1<= /span>
a2<= /span>
a3<= /span>
a4<= /span>  . ..
A nonincreasing sequence that is bounded below is a CG sequence.
------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0131_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1gEDAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQDT AQIAgAAAAAAAAAIdjI+py+0Po5y02ouz3rz7D4biSJbmiabqyrbuWAAAOw== ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0131_image047.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAwAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAC ACQAgAAAAAAAAAIHjI+py+2vCgA7 ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0131_image048.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAALvg4wECAwIYhB2nEcIrYkygSZnsdXrnPVWgg3gL8wAFADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0131_image049.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhggAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQB9 AAoAgQAAAAAAALvg4/8AAAJ7hI+pyxgNzYtp0mOv3hwHEWRXCIYUaUbhl1LD0MXQJ9TgRtuikte7 Y9NdXi+ZERMUnpK+Rk+5eDYjRCKiis1qt9yu9wsOi7mGsfmMTqvX7Lb7bT52pLcl8ydh1qN6/BUr J/Pkx5NECDCoYjhkFWi00qIBeSgBOWLJyFAAADs= ------=_NextPart_01C97CE5.F3B37830 Content-Location: file:///C:/30889913/23MonotoneSequences_files/slide0132.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Hence,
a nondecreasing sequence that is bounded above is a CG sequence.
Monotone Sequences
the upper
bound A
a1<= /span>
a2<= /span>
a3<= /span>
a4<= /span>  . ..
A nonincreasing sequence that is bounded below is a CG sequence.
the lower
bound b
a1<= /span>
a2<= /span>
a3<= /span>
...    a4 
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Hence,
a nondecreasing sequence that is bounded above is a CG sequence.
Monotone Sequences
the upper
bound A
a1<= /span>
a2<= /span>
a3<= /span>
a4<= /span>  . ..
A nonincreasing sequence that is bounded below is a CG sequence.
the lower
bound b
a1<= /span>
a2<= /span>
a3<= /span>
...    a4 
Note the theorem just gives the existence of the limit but it doesn't indicate what the limit is. 
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Example: Show {        = ;   } is a convergent sequence.
2n
1 + 3n
Monotone Sequences
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Example: Show {        = ;   } is a convergent sequence.
2n
1 + 3n
Monotone Sequences
We will show that a<= /span>n+1 – an < 0:
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Example: Show {        = ;   } is a convergent sequence.
2n
1 + 3n
Monotone Sequences
2n+1
1 + 3n+1
–=
2n