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Differ= entiation and Integration of Power Series
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Differ= entiation and Integration of Power Series
Theorem (Derivative and integra= l of Taylor series) :
------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0327.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Σ
k=3D0
∞
Differ= entiation and Integration of Power Series
Let f(x) =3D P(x) =3D the Taylor series = =3D    ck(x – a)k  for
<= /div>
all x in the interval (a – R, a + R) where R is the radius of convergence, then
Theorem (Derivative and integra= l of Taylor series) :
------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0299.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Σ
k=3D0
∞
Differ= entiation and Integration of Power Series
Let f(x) =3D P(x) =3D the Taylor series = =3D    ck(x – a)k  for
<= /div>
all x in the interval (a – R, a + R) where R is the radius of convergence, then
Theorem (Derivative and integra= l of Taylor series) :
I.  the Taylor series of f '(x) is P'(x) =3D    [ck(x – a)k]'   
with radius of convergence R and f '(x) = =3D P'(x) for
<= /div>
all x in the interval (a – R, a + R= ).
Σ
k=3D0
∞
------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0300.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Σ
k=3D0
∞
Differ= entiation and Integration of Power Series
Let f(x) =3D P(x) =3D the Taylor series = =3D    ck(x – a)k  for
<= /div>
all x in the interval (a – R, a + R) where R is the radius of convergence, then
Theorem (Derivative and integra= l of Taylor series) :
I.  the Taylor series of f '(x) is P'(x) =3D    [ck(x – a)k]'   
with radius of convergence R and f '(x) = =3D P'(x) for
<= /div>
all x in the interval (a – R, a + R= ).
Σ
k=3D0
∞
II.  the Taylor series of ∫P(x= )dx=3D   ∫ck(x – a)kdx   
with radius of convergence R and <= span lang=3DEN-US style=3D'font-family:"";mso-ascii-font-family:"";mso-fareast-= font-family: "";mso-bidi-font-family:"";mso-hansi-font-family:Arial;font-size:156%; mso-fareast-language:ZH-CN'>∫f(x)dx =3D ∫P(x)dx for
<= /div>
all x in the interval (a – R, a + R= ).
Σ
k=3D0
∞
------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0301.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Σ
k=3D0
∞
Differ= entiation and Integration of Power Series
Let f(x) =3D P(x) =3D the Taylor series = =3D    ck(x – a)k  for
<= /div>
all x in the interval (a – R, a + R) where R is the radius of convergence, then
Theorem (Derivative and integra= l of Taylor series) :
I.  the Taylor series of f '(x) is P'(x) =3D    [ck(x – a)k]'   
with radius of convergence R and f '(x) = =3D P'(x) for
<= /div>
all x in the interval (a – R, a + R= ).
Σ
k=3D0
∞
II.  the Taylor series of ∫P(x= )dx=3D   ∫ck(x – a)kdx   
with radius of convergence R and <= span lang=3DEN-US style=3D'font-family:"";mso-ascii-font-family:"";mso-fareast-= font-family: "";mso-bidi-font-family:"";mso-hansi-font-family:Arial;font-size:156%; mso-fareast-language:ZH-CN'>∫f(x)dx =3D ∫P(x)dx for
<= /div>
all x in the interval (a – R, a + R= ).
Σ
k=3D0
∞
III. ∫ f(x= )dx =3D    ∫ ck(x – a)kdx with the series converges <= /div>
<= /div>
absolutely and (c, d) is any interval in = (a – R, a + R).
Σ
k=3D0
∞
c
d
c
d
------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0301_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgAkAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAJ ACIAgAAAAP8AAAIojA2ZqMoNXpMuwGUvxXf1+G1RBokmOaFVd7Iqp72jS8d1KZ/y7OZGAQA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0301_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwAgAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAK ACAAgAAAAP8AAAIqTICJxqbJHpTrUHetxJcuD3CPqJCQtoFm5a1fi5bxCat2nY3RKTYp/igAADs= ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0295.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Given the Mac series of
Differ= entiation and Integration of Power Series
Σ
k=3D0
(2k+1)!
(-1)kx2k+1
∞
x
–
3!
x3
+
5!
x5
+ .. =3D
7!
x7
–
Σ
k=3D0
(2k)!
(-1)kx2k
∞
+
4!
x4
6!
x6
8!
x8
+
1
–
–
– .. =3D
2!
x2
sin(x) =3D
cos(x) =3D
------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0295_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA/ AAEAgAAAAAAAAAIGjI+py+1cADt= ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0295_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0295_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0295_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0295_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA/ AAEAgAAAAAAAAAIGjI+py+1cADt= ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0295_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0295_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0295_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0295_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0302.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Given the Mac series of
[sin(x)]' =3D
Differ= entiation and Integration of Power Series
Σ
k=3D0
(2k+1)!
(-1)kx2k+1
∞
x
–
3!
x3
+
5!
x5
+ .. =3D
7!
x7
–
Σ
k=3D0
(2k)!
(-1)kx2k
∞
+
4!
x4
6!
x6
8!
x8
+
1
–
–
– .. =3D
2!
x2
sin(x) =3D
cos(x) =3D
The derivative of= the power series of sin(x) is 
------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0302_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA/ AAEAgAAAAAAAAAIGjI+py+1cADt= ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0302_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0302_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0302_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0302_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA/ AAEAgAAAAAAAAAIGjI+py+1cADt= ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0302_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0302_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0302_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0302_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0303.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Given the Mac series of
[sin(x)]' =3D [  = 3;
Differ= entiation and Integration of Power Series
Σ
k=3D0
(2k+1)!
(-1)kx2k+1
∞
x
–
3!
x3
+
5!
x5
+ .. =3D
7!
x7
–
Σ
k=3D0
(2k)!
(-1)kx2k
∞
+
4!
x4
6!
x6
8!
x8
+
1
–
–
– .. =3D
2!
x2
sin(x) =3D
cos(x) =3D
x
–
3!
x3
+
5!
x5
+ .. ]'
7!
x7
–
The derivative of= the power series of sin(x) is 
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Example: Given the Mac series of
[sin(x)]' =3D [  = 3;
Differ= entiation and Integration of Power Series
Σ
k=3D0
(2k+1)!
(-1)kx2k+1
∞
x
–
3!
x3
+
5!
x5
+ .. =3D
7!
x7
–
Σ
k=3D0
(2k)!
(-1)kx2k
∞
+
4!
x4
6!
x6
8!
x8
+
1
–
–
– .. =3D
2!
x2
sin(x) =3D
cos(x) =3D
x
–
3!
x3
+
5!
x5
+ .. ]'
7!
x7
–
+
4!
x4
6!
x6
+ …
=3D 1
–
–
2!
x2
The derivative of= the power series of sin(x) is 
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file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0305.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Given the Mac series of
[sin(x)]' =3D [  = 3;
Differ= entiation and Integration of Power Series
Σ
k=3D0
(2k+1)!
(-1)kx2k+1
∞
x
–
3!
x3
+
5!
x5
+ .. =3D
7!
x7
–
Σ
k=3D0
(2k)!
(-1)kx2k
∞
+
4!
x4
6!
x6
8!
x8
+
1
–
–
– .. =3D
2!
x2
sin(x) =3D
cos(x) =3D
x
–
3!
x3
+
5!
x5
+ .. ]'
7!
x7
–
+
4!
x4
6!
x6
+ …
=3D 1
–
–
2!
x2
=3D cos(x)
The derivative of= the power series of sin(x) is 
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Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0305_image051.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0305_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA/ AAEAgAAAAAAAAAIGjI+py+1cADt= ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0305_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0305_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0305_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0305_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAY 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file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0305_image060.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0305_image061.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0305_image062.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Given the Mac series of
[sin(x)]' =3D [  = 3;
Differ= entiation and Integration of Power Series
Σ
k=3D0
(2k+1)!
(-1)kx2k+1
∞
x
–
3!
x3
+
5!
x5
+ .. =3D
7!
x7
–
Σ
k=3D0
(2k)!
(-1)kx2k
∞
+
4!
x4
6!
x6
8!
x8
+
1
–
–
– .. =3D
2!
x2
sin(x) =3D
cos(x) =3D
x
–
3!
x3
+
5!
x5
+ .. ]'
7!
x7
–
+
4!
x4
6!
x6
+ …
=3D 1
–
–
2!
x2
=3D cos(x)
The derivative of= the power series of sin(x) is 
∫sin(x)dx =3D
The antiderivativ= e of the power series of sin(x) is 
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Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image066.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image067.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA/ AAEAgAAAAAAAAAIGjI+py+1cADt= ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image068.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image069.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image070.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image071.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image072.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image073.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image074.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image075.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image076.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0306_image077.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.F5A76380 Content-Location: file:///C:/EC8AB113/36DifferentiationandIntegrationofPowerSeries_files/slide0307.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Given the Mac series of
[sin(x)]' =3D [  = 3;
Differ= entiation and Integration of Power Series
Σ
k=3D0
(2k+1)!
(-1)kx2k+1
∞
x
–
3!
x3
+
5!
x5
+ .. =3D
7!
x7
–
Σ
k=3D0
(2k)!
(-1)kx2k
∞
+
4!
x4
6!
x6
8!
x8
+
1
–
–
– .. =3D
2!
x2
sin(x) =3D
cos(x) =3D
x
–
3!
x3
+
5!
x5
+ .. ]'
7!
x7
–
+
4!
x4
6!
x6
+ …
=3D 1
–
–
2!
x2
=3D cos(x)
The derivative of= the power series of sin(x) is 
∫sin(x)dx =3D
∫= xdx
– ∫      dx=
3!
x3
+ ∫     dx
5!
x5
–…
The antiderivativ= e of the power series of sin(x) is 
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Example: Given the Mac series of
[sin(x)]' =3D [  = 3;
Differ= entiation and Integration of Power Series
Σ
k=3D0
(2k+1)!
(-1)kx2k+1
∞
x
–
3!
x3
+
5!
x5
+ .. =3D
7!
x7
–
Σ
k=3D0
(2k)!
(-1)kx2k
∞
+
4!
x4
6!
x6
8!
x8
+
1
–
–
– .. =3D
2!
x2
sin(x) =3D
cos(x) =3D
x
–
3!
x3
+
5!
x5
+ .. ]'
7!
x7
–
+
4!
x4
6!
x6
+ …
=3D 1
–
–
2!
x2
=3D cos(x)
The derivative of= the power series of sin(x) is 
∫sin(x)dx =3D
∫= xdx
– ∫      dx=
3!
x3
+ ∫     dx
5!
x5
–…
The antiderivativ= e of the power series of sin(x) is 
4!
x4
6!
x6
=3D
+
2!
x2
–
– …  + c
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