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Alter= nating Series and Conditional Convergence
------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0122.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Alter= nating Series and Conditional Convergence
An alternating series is a series with alternating positive and negative terms. 
------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0123.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Alter= nating Series and Conditional Convergence
An alternating series is a series with alternating positive and negative terms.  Alternat= ing series usally
<= /div>
are given as        = ;   
Σ
n=3D1
∞
(-1)nan =3D= -a1 + a2 – a3 + a4 ….  or as    
Σ
n=3D1
∞
(-1)n+1an =3D a1 – a2 + a3 – a4 … where an > 0
------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0125.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Alter= nating Series and Conditional Convergence
An alternating series is a series with alternating positive and negative terms.  Alternat= ing series usally
<= /div>
are given as        = ;   
Σ
n=3D1
∞
(-1)nan =3D= -a1 + a2 – a3 + a4 ….  or as    
Σ
n=3D1
∞
(-1)n+1an =3D a1 – a2 + a3 – a4 … where an > 0
From here on, we will write an alternating series as
Σ
n=3D1
∞
an where the sequence {an} h= as alternating signs.
------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0126.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Alter= nating Series and Conditional Convergence
An alternating series is a series with alternating positive and negative terms.  Alternat= ing series usally
<= /div>
are given as        = ;   
Σ
n=3D1
∞
(-1)nan =3D= -a1 + a2 – a3 + a4 ….  or as    
Σ
n=3D1
∞
(-1)n+1an =3D a1 – a2 + a3 – a4 … where an > 0
Theorem (Alternating Series): Suppose {a<= /span>n} i= s a alternating sequence and lim a<= /span>n =3D 0, and an+1 >= ; an 
<= /div>
then the alternating series        converges. 
nà∞
From here on, we will write an alternating series as
Σ
n=3D1
∞
an where the sequence {an} h= as alternating signs.
Σ
an
------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0126_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0165.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Alter= nating Series and Conditional Convergence
An alternating series is a series with alternating positive and negative terms.  Alternat= ing series usally
<= /div>
are given as        = ;   
Σ
n=3D1
∞
(-1)nan =3D= -a1 + a2 – a3 + a4 ….  or as    
Σ
n=3D1
∞
(-1)n+1an =3D a1 – a2 + a3 – a4 … where an > 0
Theorem (Alternating Series): Suppose {a<= /span>n} i= s a alternating sequence and lim a<= /span>n =3D 0, and an+1 >= ; an 
<= /div>
then the alternating series        converges. 
nà∞
From here on, we will write an alternating series as
Σ
n=3D1
∞
an where the sequence {an} h= as alternating signs.
Σ
an
We will use the alternating harmonic seri= es
<= /div>
=
to demonstrate the argument for the theorem. 
1
2
1
3
...
1
+
Σ
n=3D1
(-1)n+1      =3D
1
n
∞
1
4
 –
 –
------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0165_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0165_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0165_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0165_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0165_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0118.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Alter= nating Series and Conditional Convergence
1
2
1
3
...
1
+
Σ
n=3D1
(-1)n+1      =3D
1
n
∞
1
4
 –
 –
The convergence of 
may be demonstrated on the real line. 
------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0128.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Alter= nating Series and Conditional Convergence
1
2
1
3
...
1
+
Σ
n=3D1
(-1)n+1      =3D
1
n
∞
1
4
 –
 –
The convergence of 
may be demonstrated on the real line. 
Let Sn =3D=         = ;            &n= bsp;            = ;   be the partial sum.  <= /span>
1
2
1
3
...
1
+
1
4
 –
 –
1
n
&pl= usmn;
------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Alter= nating Series and Conditional Convergence
1
2
1
3
...
1
+
Σ
n=3D1
(-1)n+1      =3D
1
n
∞
1
4
 –
 –
The convergence of 
may be demonstrated on the real line. 
Let Sn =3D=         = ;            &n= bsp;            = ;   be the partial sum.  <= /span>
1
2
1
3
...
1
+
1
4
 –
 –
1
n
&pl= usmn;
We may visualize Sn geometrically as following: 
0
1
S1
1
------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhxQEMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADB AQoAgAAAAAAAAAKAhI+py+0Po5y02ouz3rz7FnziSJYmEp7qyrbuG6XwTNexjef6zsr8GPD9bMKh 8YgcFpOZ4JJ5ekKn1CpHao05Ac6u9wsOi8fksvmMTqvX7Lb7DY/L5/S6XZ0tYfP8Pn/vt7AV6AFI eIioY3g4mLix6BgpGTWp0FhpAYm5ydnpUQAAOw== ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwELAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwAP AQUAgAAAAP8AAAI6hI+py+0Po5y02ouzjmH7D4biSJZhgBroyrbuC8fyTNf2jef6zvf+DwwKh8Si 8Yi0mZbMpvMJ1aQMBQA7 ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0129_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAoAgQAAAAAAAP8AAAECAwIWBIJoJ93BnENyhvoupjtKeFzJYoBGAQA7 ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0130.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Alter= nating Series and Conditional Convergence
1
2
1
3
...
1
+
Σ
n=3D1
(-1)n+1      =3D
1
n
∞
1
4
 –
 –
The convergence of 
may be demonstrated on the real line. 
Let Sn =3D=         = ;            &n= bsp;            = ;   be the partial sum.  <= /span>
1
2
1
3
...
1
+
1
4
 –
 –
1
n
&pl= usmn;
We may visualize Sn geometrically as following: 
0
1
S1
1
S2
1/2
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Alter= nating Series and Conditional Convergence
1
2
1
3
...
1
+
Σ
n=3D1
(-1)n+1      =3D
1
n
∞
1
4
 –
 –
The convergence of 
may be demonstrated on the real line. 
Let Sn =3D=         = ;            &n= bsp;            = ;   be the partial sum.  <= /span>
1
2
1
3
...
1
+
1
4
 –
 –
1
n
&pl= usmn;
We may visualize Sn geometrically as following: 
0
1
S1
1
S2
1/2
S3
1/3
------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0131.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0131_image031.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0131_image032.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0131_image033.gif 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R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0131_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0131_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0131_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhxQEMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADB 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file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0131_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAoAgQAAAAAAAP8AAAECAwIWBIJoJ93BnENyhvoupjtKeFzJYoBGAQA7 ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0131_image043.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAK AAkAgQAAAAAAAP8AAAECAwIXBIJoFuIe2JuxTXjuQ1rUHmXUciQhUAAAOw== ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0131_image044.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhWQALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwBU AAYAgAAAAP8AAAIjhI+py+0PQ4C02hvlBLr7D4biSJbmiaaqirXum2jwTFvbUQAAOw== ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0131_image045.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAK AAoAgQAAAAAAAP8AAAECAwIYhINoJ93BnINBzmofwJZmQXHagoDQQqIFADs= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Alter= nating Series and Conditional Convergence
1
2
1
3
...
1
+
Σ
n=3D1
(-1)n+1      =3D
1
n
∞
1
4
 –
 –
The convergence of 
may be demonstrated on the real line. 
Let Sn =3D=         = ;            &n= bsp;            = ;   be the partial sum.  <= /span>
1
2
1
3
...
1
+
1
4
 –
 –
1
n
&pl= usmn;
We may visualize Sn geometrically as following: 
0
1
S1
1
S2
1/2
S3
1/3
S4
1/4
------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image047.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image048.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image049.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image050.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image051.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhxQEMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADB AQoAgAAAAAAAAAKAhI+py+0Po5y02ouz3rz7FnziSJYmEp7qyrbuG6XwTNexjef6zsr8GPD9bMKh 8YgcFpOZ4JJ5ekKn1CpHao05Ac6u9wsOi8fksvmMTqvX7Lb7DY/L5/S6XZ0tYfP8Pn/vt7AV6AFI eIioY3g4mLix6BgpGTWp0FhpAYm5ydnpUQAAOw== ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwELAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwAP AQUAgAAAAP8AAAI6hI+py+0Po5y02ouzjmH7D4biSJZhgBroyrbuC8fyTNf2jef6zvf+DwwKh8Si 8Yi0mZbMpvMJ1aQMBQA7 ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhiQALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwCG AAUAgAAAAP8AAAInhBFxy+0Po5y0WprS3bz7v2XiSJbmiabqyraugmTgTNf2pN36zlMFADs= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image057.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAoAgQAAAAAAAP8AAAECAwIWBIJoJ93BnENyhvoupjtKeFzJYoBGAQA7 ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image058.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAK AAkAgQAAAAAAAP8AAAECAwIXBIJoFuIe2JuxTXjuQ1rUHmXUciQhUAAAOw== ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0132_image059.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhWQALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwBU AAYAgAAAAP8AAAIjhI+py+0PQ4C02hvlBLr7D4biSJbmiaaqirXum2jwTFvbUQAAOw== 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Alter= nating Series and Conditional Convergence
1
2
1
3
...
1
+
Σ
n=3D1
(-1)n+1      =3D
1
n
∞
1
4
 –
 –
The convergence of 
may be demonstrated on the real line. 
Let Sn =3D=         = ;            &n= bsp;            = ;   be the partial sum.  <= /span>
1
2
1
3
...
1
+
1
4
 –
 –
1
n
&pl= usmn;
We may visualize Sn geometrically as following: 
0
1
S1
1
S2
1/2
S3
1/3
S4
1/4
=
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Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0133_image065.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0133_image066.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C9C737.A8E49090 Content-Location: file:///C:/2AA444C5/28AlternatingSeriesandConditionalConvergence_files/slide0133_image067.gif Content-Transfer-Encoding: base64 Content-Type: image/gif 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Alter= nating Series and Conditional Convergence
1
2
1
3
...
1
+
Σ
n=3D1
(-1)n+1      =3D
1
n
∞
1
4
 –
 –
The convergence of 
may be demonstrated on the real line. 
Let Sn =3D=         = ;            &n= bsp;            = ;   be the partial sum.  <= /span>
1
2
1
3
...
1
+
1
4
 –
 –
1
n
&pl= usmn;
We may visualize Sn geometrically as following: 
0
1
S1
1
S2
1/2
S3
1/3
S4
1/4
=
Since     à 0, the red dot must approach a limit
<= /div>
which is=  
1
n
Σ
n=3D1
(-1)n+1    .
1
n
∞
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Alter= nating Series and Conditional Convergence
1
2
1
3
...
1
+
Σ
n=3D1
(-1)n+1      =3D
1
n
∞
1
4
 –
 –
The convergence of 
may be demonstrated on the real line. 
Let Sn =3D=         = ;            &n= bsp;            = ;   be the partial sum.  <= /span>
1
2
1
3
...
1
+
1
4
 –
 –
1
n
&pl= usmn;
We may visualize Sn geometrically as following: 
0
1
S1
1
S2
1/2
S3
1/3
S4
1/4
=
Since     à 0, the red dot must approach a limit
<= /div>
which is=  
1
n
Σ
n=3D1
(-1)n+1    .
1
n
∞
(Later we'll see its Ln(2) » 0.59)
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