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------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/master03.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAAD/wAAAAYATwAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwC7wLpAwMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAMAAAAQAkhAPAAAAAAAAAA 7wLpA/////8IAAAA+gIAAAAAAAAAAAAABAAAAC0BAgAHAAAA/AIAAP///wAAAAQAAAAtAQMABAAA ACcB//8cAAAA+wLZ/wAAAAAAAJABAAAAAABAACJBcmlhbAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAQAAAAtAQQABAAAAC4BGAAEAAAAAgEBAAUAAAAJAgAAAAJPAAAAMgosAEAAMAAAAENvbXB1 dGF0aW9uIFRlY2huaXF1ZXMgZm9yIE1hY2xhdXJpbiBFeHBhbnNpb25zIBwAFgAhABYAFQALABYA CwAIABYAFgALABcAFgATABYAFgAIABYAFgAWABMACgAMABYADAALACEAFQAUAAgAFgAWAA0ACAAW AAsAGgASABYAFgAWABMACQAWABYAEwALAAQAAAAuAQAAHAAAAPsCEAAHAAAAAAC8AgAAAAABAgIi U3lzdGVtAAAAAAAAGAAAAGhpEwACAAAA5AQAAAAAAAAEAAAALQEFAAQAAADwAQQAAwAAAAAA ------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0094.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Comput= ation Techniques for Maclaurin Expansions
------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0211.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Comput= ation Techniques for Maclaurin Expansions
Direct computation of the Mac-series can be messy=
via derivatives.
------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0215.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Comput= ation Techniques for Maclaurin Expansions
Direct computation of the Mac-series can be messy=
via derivatives. In this section we show some of = the
algebraic techniques for computing the Mac-series= .
------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0216.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Comput= ation Techniques for Maclaurin Expansions
Direct computation of the Mac-series can be messy=
via derivatives. In this section we show some of = the
algebraic techniques for computing the Mac-series= .
Theorem:  Let F(x) and G(x) be the Mac-series of
f(x) and and g(x) respectively.  
------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0217.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Comput= ation Techniques for Maclaurin Expansions
Direct computation of the Mac-series can be messy=
via derivatives. In this section we show some of = the
algebraic techniques for computing the Mac-series= .
Theorem:  Let F(x) and G(x) be the Mac-series of
f(x) and and g(x) respectively.  
I. The Mac-expansions respect +, –, * , and= /, that is,
the Mac-series of f + g, f – g, f*g, and f/g= are
F + G, F – G, F*G, and F/G respectively.= 3;
------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0218.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Comput= ation Techniques for Maclaurin Expansions
Direct computation of the Mac-series can be messy=
via derivatives. In this section we show some of = the
algebraic techniques for computing the Mac-series= .
Theorem:  Let F(x) and G(x) be the Mac-series of
f(x) and and g(x) respectively.  
I. The Mac-expansions respect +, –, * , and= /, that is,
the Mac-series of f + g, f – g, f*g, and f/g= are
F + G, F – G, F*G, and F/G respectively.= 3;
II. Mac-series respect composition of functions.&= #13;
This is particularly useful if g(x) is a polynomi= al in
which case the Mac-series of f(g(x)) is F(g(x)).<= /span>
------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0219.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Comput= ation Techniques for Maclaurin Expansions
Direct computation of the Mac-series can be messy=
via derivatives. In this section we show some of = the
algebraic techniques for computing the Mac-series= .
Theorem:  Let F(x) and G(x) be the Mac-series of
f(x) and and g(x) respectively.  
I. The Mac-expansions respect +, –, * , and= /, that is,
the Mac-series of f + g, f – g, f*g, and f/g= are
F + G, F – G, F*G, and F/G respectively.= 3;
II. Mac-series respect composition of functions.&= #13;
This is particularly useful if g(x) is a polynomi= al in
which case the Mac-series of f(g(x)) is F(g(x)).<= /span>
We list below the basic Mac-series that we will u= se
in our examples . 
------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0210.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P.
Mac-series of polynomials are themselves.=
II. For ex, its=
Σ
k=3D0
k! .
xk=
∞
x
+
2!
1
+
x2=
+ ..
+
+
3!
x3=
n! ..<= /span>
xn=
=3D
Σ
k=3D0
(2k+1= )!
(-1)<= /span>k= x2k+1
∞
x
–= ;
3!
x3=
+
5!
x5=
+ .. = =3D
7!
x7=
–= ;
III. For sin(x), its=
IV. For cos(x), its=
Σ
k=3D0
(2k)!<= /span>
(-1)<= /span>k= x2k
∞
+
4!
x4=
6!
x6=
8!
x8=
+
1
–= ;
–= ;
̵= 1; .. =3D
2!
x2=
V. For        = ;     , its
(1 &#= 8211; x )
1
1 + x + x2 + x3 + x4 .. =3D
Σ
k=3D0
∞
xk=
Comput= ation Techniques for Maclaurin Expansions
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Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P.
Mac-series of polynomials are themselves.=
II. For ex, its=
Σ
k=3D0
k! .
xk=
∞
x
+
2!
1
+
x2=
+ ..
+
+
3!
x3=
n! ..<= /span>
xn=
=3D
Σ
k=3D0
(2k+1= )!
(-1)<= /span>k= x2k+1
∞
x
–= ;
3!
x3=
+
5!
x5=
+ .. = =3D
7!
x7=
–= ;
III. For sin(x), its=
IV. For cos(x), its=
Σ
k=3D0
(2k)!<= /span>
(-1)<= /span>k= x2k
∞
+
4!
x4=
6!
x6=
8!
x8=
+
1
–= ;
–= ;
̵= 1; .. =3D
2!
x2=
V. For        = ;     , its
(1 &#= 8211; x )
1
1 + x + x2 + x3 + x4 .. =3D
Σ
k=3D0
∞
xk=
Comput= ation Techniques for Maclaurin Expansions
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Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P.
Mac-series of polynomials are themselves.=
II. For ex, its=
Σ
k=3D0
k! .
xk=
∞
x
+
2!
1
+
x2=
+ ..
+
+
3!
x3=
n! ..<= /span>
xn=
=3D
Σ
k=3D0
(2k+1= )!
(-1)<= /span>k= x2k+1
∞
x
–= ;
3!
x3=
+
5!
x5=
+ .. = =3D
7!
x7=
–= ;
III. For sin(x), its=
IV. For cos(x), its=
Σ
k=3D0
(2k)!<= /span>
(-1)<= /span>k= x2k
∞
+
4!
x4=
6!
x6=
8!
x8=
+
1
–= ;
–= ;
̵= 1; .. =3D
2!
x2=
V. For        = ;     , its
(1 &#= 8211; x )
1
1 + x + x2 + x3 + x4 .. =3D
Σ
k=3D0
∞
xk=
Comput= ation Techniques for Maclaurin Expansions
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Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P.
Mac-series of polynomials are themselves.=
II. For ex, its=
Σ
k=3D0
k! .
xk=
∞
x
+
2!
1
+
x2=
+ ..
+
+
3!
x3=
n! ..<= /span>
xn=
=3D
Σ
k=3D0
(2k+1= )!
(-1)<= /span>k= x2k+1
∞
x
–= ;
3!
x3=
+
5!
x5=
+ .. = =3D
7!
x7=
–= ;
III. For sin(x), its=
IV. For cos(x), its=
Σ
k=3D0
(2k)!<= /span>
(-1)<= /span>k= x2k
∞
+
4!
x4=
6!
x6=
8!
x8=
+
1
–= ;
–= ;
̵= 1; .. =3D
2!
x2=
V. For        = ;     , its
(1 &#= 8211; x )
1
1 + x + x2 + x3 + x4 .. =3D
Σ
k=3D0
∞
xk=
Comput= ation Techniques for Maclaurin Expansions
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Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P.
Mac-series of polynomials are themselves.=
II. For ex, its=
Σ
k=3D0
k! .
xk=
∞
x
+
2!
1
+
x2=
+ ..
+
+
3!
x3=
n! ..<= /span>
xn=
=3D
Σ
k=3D0
(2k+1= )!
(-1)<= /span>k= x2k+1
∞
x
–= ;
3!
x3=
+
5!
x5=
+ .. = =3D
7!
x7=
–= ;
III. For sin(x), its=
IV. For cos(x), its=
Σ
k=3D0
(2k)!<= /span>
(-1)<= /span>k= x2k
∞
+
4!
x4=
6!
x6=
8!
x8=
+
1
–= ;
–= ;
̵= 1; .. =3D
2!
x2=
V. For        = ;     , its
(1 &#= 8211; x )
1
1 + x + x2 + x3 + x4 .. =3D
Σ
k=3D0
∞
xk=
Comput= ation Techniques for Maclaurin Expansions
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Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P.
Mac-series of polynomials are themselves.=
II. For ex, its=
Σ
k=3D0
k! .
xk=
∞
x
+
2!
1
+
x2=
+ ..
+
+
3!
x3=
n! ..<= /span>
xn=
=3D
Σ
k=3D0
(2k+1= )!
(-1)<= /span>k= x2k+1
∞
x
–= ;
3!
x3=
+
5!
x5=
+ .. = =3D
7!
x7=
–= ;
III. For sin(x), its=
IV. For cos(x), its=
Σ
k=3D0
(2k)!<= /span>
(-1)<= /span>k= x2k
∞
+
4!
x4=
6!
x6=
8!
x8=
+
1
–= ;
–= ;
̵= 1; .. =3D
2!
x2=
V. For        = ;     , its
(1 &#= 8211; x )
1
1 + x + x2 + x3 + x4 .. =3D
Σ
k=3D0
∞
xk=
Comput= ation Techniques for Maclaurin Expansions
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Example: Find the Mac-series of sin(x) + cos(x)
Comput= ation Techniques for Maclaurin Expansions
------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0225.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Find the Mac-series of sin(x) + cos(x)
Σ
k=3D0
(2k+1)!
(-1)kx2k+1
∞
x
–
3!
x3
+
5!
x5
+ .. =3D
7!
x7
–
sin(x) =3D=
Comput= ation Techniques for Maclaurin Expansions
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Example: Find the Mac-series of sin(x) + cos(x)
Σ
k=3D0
(2k+1)!
(-1)kx2k+1
∞
x
–
3!
x3
+
5!
x5
+ .. =3D
7!
x7
–
sin(x) =3D=
cos(x) =3D=
Σ
k=3D0
(2k)!
(-1)kx2k
∞
+
4!
x4
6!
x6
8!
x8
+
1
–
–
– .. =3D
2!
x2
Comput= ation Techniques for Maclaurin Expansions
------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0226_image092.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA/ AAEAgAAAAAAAAAIGjI+py+1cADt= ------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0226_image093.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhfAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAB6 AAIAgAAAAAAAAAIVhI+py30Bo5w0upuq3rxDDIbiSGIFADs= ------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0226_image094.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA/ AAEAgAAAAAAAAAIGjI+py+1cADt= ------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0226_image095.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhfAAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAB7 AAMAgAAAAAAAAAIZjI+pB+0PI1yUyouz3rz7D4biSJYmWKVJAQA7 ------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0226_image096.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.B221C650 Content-Location: file:///C:/23664A33/32ComputationTechniquesforMacLaurinExpansions_files/slide0227.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Find the Mac-series of sin(x) + cos(x)
Σ
k=3D0
(2k+1)!
(-1)kx2k+1
∞
x
–
3!
x3
+
5!
x5
+ .. =3D
7!
x7
–
sin(x) =3D=
cos(x) =3D=
Σ
k=3D0
(2k)!
(-1)kx2k
∞
+
4!
x4
6!
x6
8!
x8
+
1
–
–
– .. =3D
2!
x2
Comput= ation Techniques for Maclaurin Expansions
Therefore,
sin(x) + cos(x) = =3D
1
+
x
–
2!
x2
–
3!
x3
+
4!
x4
+
5!
x5
6!
x6