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Power Series 
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A <= /span>power series P(x) is a "polynomial&quo= t; in x of infinitely many terms.
Power Series 
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A <= /span>power series P(x) is a "polynomial&quo= t; in x of infinitely many terms. That is,
Σ
n=3D0
∞
anxn =3D a0 + a1x += a2x2 + a3x3 + a4x4 ….  
Power Series 
P(x) =3D  
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A <= /span>power series P(x) is a "polynomial&quo= t; in x of infinitely many terms. That is,
Σ
n=3D0
∞
anxn =3D a0 + a1x += a2x2 + a3x3 + a4x4 ….  
Note that P(0) =3D a0.
Power Series 
P(x) =3D  
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A <= /span>power series P(x) is a "polynomial&quo= t; in x of infinitely many terms. That is,
Σ
n=3D0
∞
anxn =3D a0 + a1x += a2x2 + a3x3 + a4x4 ….  
The basic question for a power series is to determine <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>the value(s) of x where it converges or diverges.
Note that P(0) =3D a0.
Power Series 
P(x) =3D  
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0169.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
A <= /span>power series P(x) is a "polynomial&quo= t; in x of infinitely many terms. That is,
Σ
n=3D0
∞
anxn =3D a0 + a1x += a2x2 + a3x3 + a4x4 ….  
Example: A.  Let P(x) =3D 1 + x + x2 + x3 + x4 …
The basic question for a power series is to determine <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>the value(s) of x where it converges or diverges.
Note that P(0) =3D a0.
Power Series 
P(x) =3D  
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0170.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
A <= /span>power series P(x) is a "polynomial&quo= t; in x of infinitely many terms. That is,
Σ
n=3D0
∞
anxn =3D a0 + a1x += a2x2 + a3x3 + a4x4 ….  
Example: A.  Let P(x) =3D 1 + x + x2 + x3 + x4 …
From the theorem of geometr= ic series, given x =3D r,
I. P(r) =3D 1 + r + r2 + r3 + r4 + … converges i= f | r | < 1.
The basic question for a power series is to determine <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>the value(s) of x where it converges or diverges.
Note that P(0) =3D a0.
Power Series 
P(x) =3D  
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A <= /span>power series P(x) is a "polynomial&quo= t; in x of infinitely many terms. That is,
Σ
n=3D0
∞
anxn =3D a0 + a1x += a2x2 + a3x3 + a4x4 ….  
Example: A.  Let P(x) =3D 1 + x + x2 + x3 + x4 …
From the theorem of geometr= ic series, given x =3D r,
I. P(r) =3D 1 + r + r2 + r3 + r4 + … converges i= f | r | < 1.
II.  P(r) diverges if | r | > 1.
The basic question for a power series is to determine <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>the value(s) of x where it converges or diverges.
Note that P(0) =3D a0.
Power Series 
P(x) =3D  
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0171_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0172.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
A <= /span>power series P(x) is a "polynomial&quo= t; in x of infinitely many terms. That is,
Σ
n=3D0
∞
anxn =3D a0 + a1x += a2x2 + a3x3 + a4x4 ….  
Example: A.  Let P(x) =3D 1 + x + x2 + x3 + x4 …
From the theorem of geometr= ic series, given x =3D r,
I. P(r) =3D 1 + r + r2 + r3 + r4 + … converges i= f | r | < 1.
II.  P(r) diverges if | r | > 1.
The basic question for a power series is to determine <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>the value(s) of x where it converges or diverges.
Note that P(0) =3D a0.
Power Series 
P(x) =3D  
0
-1
converges (absolutely)
1
diverges
diverges
P(x= ) =3D Σ
xn
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Example: B. Let P(x) =3D 1 +
Power Series 
x
x2
+
2
+
x3
3
+
x4
4
+..
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0160_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0160_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0160_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0173.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: B. Let P(x) =3D 1 +
Use the ceiling theorem:
Power Series 
x
x2
+
2
+
x3
3
+
x4
4
+..
P(x) =3D 1 +
|x|
|x|2
+
2
+
3
+
4
+..
|x|3
|x|4
1 +|x| + |x2| + |x3|  + |x4| +.. …
vI
vI
vI
vI
vI
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Example: B. Let P(x) =3D 1 +
Use the ceiling theorem:
Power Series 
x
x2
+
2
+
x3
3
+
x4
4
+..
P(x) =3D 1 +
|x|
|x|2
+
2
+
3
+
4
+..
|x|3
|x|4
1 +|x| + |x2| + |x3|  + |x4| +.. …
vI
vI
vI
vI
vI
and 1 +|x| + |x2| + |x3|  + |x4| +… converges for | x | < 1,
we conclude that
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Example: B. Let P(x) =3D 1 +
Use the ceiling theorem:
Power Series 
x
x2
+
2
+
x3
3
+
x4
4
+..
P(x) =3D 1 +
|x|
|x|2
+
2
+
3
+
4
+..
|x|3
|x|4
1 +|x| + |x2| + |x3|  + |x4| +.. …
vI
vI
vI
vI
vI
and 1 +|x| + |x2| + |x3|  + |x4| +… converges for | x | < 1,
we conclude that
P(x) =3D 1 +
|x|
|x|2
+
2
+
3
+
4
+..
|x|3
|x|4
also converges absolutely for |x| < 1.
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0175.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0175_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0175_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0175_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0175_image027.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0175_image028.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0175_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0175_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0175_image031.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0175_image032.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0161.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Power Series 
For x =3D = ±1:
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0176.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
P(1) =3D 1 +        = ;            &n= bsp;            is basically the
harmonic series, so it diverges.
Power Series 
1
1
+
2
+
1
3
+
1
4
+..
For x =3D = ±1:
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0176_image033.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0176_image034.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0176_image035.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0177.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
P(1) =3D 1 +        = ;            &n= bsp;            is basically the
harmonic series, so it diverges.
Power Series 
1
1
+
2
+
1
3
+
1
4
+..
P(-1) =3D 1 –        = ;            &n= bsp;             is basically the  = ;
alternaing harmonic series.= It converges conditionally.
1
1
–
2
+
1
3
+
1
4
– ..
For x =3D = ±1:
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0177_image036.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0177_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0177_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0177_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0177_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0177_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0178.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
P(1) =3D 1 +        = ;            &n= bsp;            is basically the
harmonic series, so it diverges.
Power Series 
1
1
+
2
+
1
3
+
1
4
+..
For x =3D | r | &= gt; 1,
P(-1) =3D 1 –        = ;            &n= bsp;             is basically the  = ;
alternaing harmonic series.= It converges conditionally.
1
1
–
2
+
1
3
+
1
4
– ..
For x =3D = ±1:
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0178_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0178_image043.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0178_image044.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0178_image045.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0178_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0178_image047.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0179.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
P(1) =3D 1 +        = ;            &n= bsp;            is basically the
harmonic series, so it diverges.
Power Series 
1
1
+
2
+
1
3
+
1
4
+..
For x =3D | r | &= gt; 1, the general terms of P(r)
P(-1) =3D 1 –        = ;            &n= bsp;             is basically the  = ;
alternaing harmonic series.= It converges conditionally.
1
1
–
2
+
1
3
+
1
4
– ..
For x =3D = ±1:
rn
n
W= 34;
as n à ∞&nbs= p; =
à
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0179_image048.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0179_image049.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0179_image050.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0179_image051.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0179_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0179_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0179_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0180.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
P(1) =3D 1 +        = ;            &n= bsp;            is basically the
harmonic series, so it diverges.
Power Series 
1
1
+
2
+
1
3
+
1
4
+..
For x =3D | r | &= gt; 1, the general terms of P(r)
P(-1) =3D 1 –        = ;            &n= bsp;             is basically the  = ;
alternaing harmonic series.= It converges conditionally.
1
1
–
2
+
1
3
+
1
4
– ..
For x =3D = ±1:
rn
n
W= 34;
as n à ∞&nbs= p; (verify this by L'Hopital's rule).
à
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0180_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0180_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0180_image057.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0180_image058.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0180_image059.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0180_image060.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0180_image061.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0181.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
P(1) =3D 1 +        = ;            &n= bsp;            is basically the
harmonic series, so it diverges.
Power Series 
1
1
+
2
+
1
3
+
1
4
+..
For x =3D | r | &= gt; 1, the general terms of P(r)
P(r) =3D 1 +
r
r2
+
2
+
3
+
4
+.. diverges.
r3
r4
P(-1) =3D 1 –        = ;            &n= bsp;             is basically the  = ;
alternaing harmonic series.= It converges conditionally.
1
1
–
2
+
1
3
+
1
4
– ..
For x =3D = ±1:
rn
n
W= 34;
as n à ∞&nbs= p; (verify this by L'Hopital's rule).
à
Hence
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P(1) =3D 1 +        = ;            &n= bsp;            is basically the
harmonic series, so it diverges.
Power Series 
1
1
+
2
+
1
3
+
1
4
+..
For x =3D | r | &= gt; 1, the general terms of P(r)
P(r) =3D 1 +
r
r2
+
2
+
3
+
4
+.. diverges.
r3
r4
P(-1) =3D 1 –        = ;            &n= bsp;             is basically the  = ;
alternaing harmonic series.= It converges conditionally.
1
1
–
2
+
1
3
+
1
4
– ..
For x =3D = ±1:
rn
n
W= 34;
as n à ∞&nbs= p; (verify this by L'Hopital's rule).
à
Hence
We summarize the result using the following picture:
------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0182.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0182_image073.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0182_image074.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0182_image075.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0182_image076.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0182_image077.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0182_image078.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0182_image079.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0182_image080.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0182_image081.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0182_image082.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.73BCB8C0 Content-Location: file:///C:/EC8AB113/29PowerSeries_files/slide0184.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
P(1) =3D 1 +        = ;            &n= bsp;            is basically the
harmonic series, so it diverges.
Power Series 
1
1
+
2
+
1
3
+
1
4
+..
For x =3D | r | &= gt; 1, the general terms of P(r)
P(r) =3D 1 +
r
r2
+
2
+
3
+
4
+.. diverges.
r3
r4
P(-1) =3D 1 –        = ;            &n= bsp;             is basically the  = ;
alternaing harmonic series.= It converges conditionally.
1
1
–
2
+
1
3
+
1
4
– ..
For x =3D = ±1:
rn
n
W= 34;
as n à ∞&nbs= p; (verify this by L'Hopital's rule).
à
Hence
We summarize the result using the following picture:
0
-1=
converges (absolutely)
1
=
diverges
diverges
P(x)= =3D Σ
xn
n
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