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5pt'><span
lang=3DEN-US>I4 =3Dlog(1+x)</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman"'>从这个例子可以=
看出,</span><span
lang=3DEN-US>symsum()</span><span style=3D'font-family:SimSun;mso-ascii-fon=
t-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>这个=
函数不但可以处理常=
5968;项级数,也可以处ச=
2;函数项级数。</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman"'>(二)函数的泰=
勒展开</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman"'>级数是高等数学=
中函数的一种重要表=
1034;形式,有许多复杂௚=
0;函数都可以用级数简&=
#21333;地来表示,而将一=
010;复杂的函数展开成幂=
;级数并取其前面的若&#=
24178;项来近似表达这个Ñ=
89;数是一种很好的近似=
方法,在学习级数的=
6102;候,我们知道将一ߑ=
0;函数展开成级数有时&=
#26159;比较麻烦的,现在=
105;们介绍用</span><span
lang=3DEN-US>Matlab</span><span style=3D'font-family:SimSun;mso-ascii-font-=
family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>展开=
函数的方法。</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman"'>在</span><span lang=3DEN-US>Matlab</span><span
style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman"'>中,用于幂级数=
展开的函数为</span><span
lang=3DEN-US>taylor()</span><span style=3D'font-family:SimSun;mso-ascii-fon=
t-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>,其=
具体格式为:</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
lang=3DEN-US><span
style=3D'mso-spacerun:yes'> =
&nb=
sp;
</span><st1:place w:st=3D"on"><st1:City w:st=3D"on">taylor</st1:City></st1:=
place>(function,n,x,a)</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
lang=3DEN-US>function</span><span style=3D'font-family:SimSun;mso-ascii-fon=
t-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>是待=
展开的函数表达式,</s=
pan><span
lang=3DEN-US>n</span><span style=3D'font-family:SimSun;mso-ascii-font-famil=
y:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>为展开项&#=
25968;,缺省是展开至</span><span
lang=3DEN-US>5</span><span style=3D'font-family:SimSun;mso-ascii-font-famil=
y:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>次幂,即</=
span><span
lang=3DEN-US>6</span><span style=3D'font-family:SimSun;mso-ascii-font-famil=
y:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>项,</span><span
lang=3DEN-US>v</span><span style=3D'font-family:SimSun;mso-ascii-font-famil=
y:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>是</span><span lang=3DEN-US>=
function</span><span
style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman"'>中的变量,</span><span
lang=3DEN-US>a</span><span style=3D'font-family:SimSun;mso-ascii-font-famil=
y:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>为函数的&#=
23637;开点,缺省为</span><span
lang=3DEN-US>0</span><span style=3D'font-family:SimSun;mso-ascii-font-famil=
y:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>,即麦克&#=
21171;林展开。</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman"'>例</span><span lang=3DEN-US>3</span><span
style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman"'>:将函数</span><span lang=3DEN-U=
S><sub><!--[if gte vml 1]><v:shape
id=3D"_x0000_i1031" type=3D"#_x0000_t75" style=3D'width:26.25pt;height:14.=
25pt'
o:ole=3D"">
<v:imagedata src=3D"sy12.files/image013.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D35 height=3D19
src=3D"sy12.files/image014.gif" v:shapes=3D"_x0000_i1031"><![endif]></sub><=
!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
031"
DrawAspect=3D"Content" ObjectID=3D"_1220424314">
</o:OLEObject>
</xml><![endif]--></span><span style=3D'font-family:SimSun;mso-ascii-font-f=
amily:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>展开=
为</span><span
lang=3DEN-US><sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1032" type=3D"#=
_x0000_t75"
style=3D'width:9.75pt;height:11.25pt' o:ole=3D"">
<v:imagedata src=3D"sy12.files/image015.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D13 height=3D15
src=3D"sy12.files/image016.gif" v:shapes=3D"_x0000_i1032"><![endif]></sub><=
!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
032"
DrawAspect=3D"Content" ObjectID=3D"_1220424315">
</o:OLEObject>
</xml><![endif]--></span><span style=3D'font-family:SimSun;mso-ascii-font-f=
amily:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>的幂=
级数,分别展开至</span><span
lang=3DEN-US>5</span><span style=3D'font-family:SimSun;mso-ascii-font-famil=
y:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>次和</span><span
lang=3DEN-US>20</span><span style=3D'font-family:SimSun;mso-ascii-font-fami=
ly:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>次。</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left'><span style=3D'=
font-family:
SimSun;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times=
New Roman"'>解:</span><span
lang=3DEN-US>Matlab</span><span style=3D'font-family:SimSun;mso-ascii-font-=
family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>程序=
为:</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
lang=3DEN-US>clear</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
lang=3DEN-US>syms x</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
lang=3DEN-US>f=3Dsin(x);</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><st1:place
w:st=3D"on"><st1:City w:st=3D"on"><span lang=3DEN-US>taylor</span></st1:Cit=
y></st1:place><span
lang=3DEN-US>(f)</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><st1:place
w:st=3D"on"><st1:City w:st=3D"on"><span lang=3DEN-US>taylor</span></st1:Cit=
y></st1:place><span
lang=3DEN-US>(f,20)</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left'><span style=3D'=
font-family:
SimSun;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times=
New Roman"'>结果为:</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:10.5=
pt'><span
lang=3DEN-US>ans =3Dx-1/6*x^3+1/120*x^5</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'margin-left:31.5pt;text-align:le=
ft;
text-indent:-21.0pt'><span lang=3DEN-US>ans=3Dx-1/6*x^3+1/120*x^5-1/5040*x^=
7+1/362880*x^9-1/39916800*x^11+1/6227020800*x^13-1/1307674368000*x^15+1/355=
687428096000*x^17-1/121645100408832000*x^19</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman"'>例</span><span lang=3DEN-US>4</span><span
style=3D'font-family:SimSun;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman"'>:将函数</span><span lang=3DEN-U=
S><sub><!--[if gte vml 1]><v:shape
id=3D"_x0000_i1033" type=3D"#_x0000_t75" style=3D'width:38.25pt;height:18p=
t' o:ole=3D"">
<v:imagedata src=3D"sy12.files/image017.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D51 height=3D24
src=3D"sy12.files/image018.gif" v:shapes=3D"_x0000_i1033"><![endif]></sub><=
!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
033"
DrawAspect=3D"Content" ObjectID=3D"_1220424317">
</o:OLEObject>
</xml><![endif]--></span><span style=3D'font-family:SimSun;mso-ascii-font-f=
amily:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>展开=
为</span><span
lang=3DEN-US><sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1034" type=3D"#=
_x0000_t75"
style=3D'width:9.75pt;height:11.25pt' o:ole=3D"">
<v:imagedata src=3D"sy12.files/image019.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D13 height=3D15
src=3D"sy12.files/image016.gif" v:shapes=3D"_x0000_i1034"><![endif]></sub><=
!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
034"
DrawAspect=3D"Content" ObjectID=3D"_1220424318">
</o:OLEObject>
</xml><![endif]--></span><span style=3D'font-family:SimSun;mso-ascii-font-f=
amily:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>的幂=
级数,</span><span
lang=3DEN-US><sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1035" type=3D"#=
_x0000_t75"
style=3D'width:12.75pt;height:11.25pt' o:ole=3D"">
<v:imagedata src=3D"sy12.files/image020.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D17 height=3D15
src=3D"sy12.files/image021.gif" v:shapes=3D"_x0000_i1035"><![endif]></sub><=
!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
035"
DrawAspect=3D"Content" ObjectID=3D"_1220424319">
</o:OLEObject>
</xml><![endif]--></span><span style=3D'font-family:SimSun;mso-ascii-font-f=
amily:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>为任=
意常数。展开至</span><span
lang=3DEN-US>4</span><span style=3D'font-family:SimSun;mso-ascii-font-famil=
y:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>次幂。</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left'><span style=3D'=
font-family:
SimSun;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times=
New Roman"'>解:</span><span
lang=3DEN-US>Matlab</span><span style=3D'font-family:SimSun;mso-ascii-font-=
family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>程序=
为:</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
lang=3DEN-US>clear</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
lang=3DEN-US>syms x m</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><span
lang=3DEN-US>f=3D(1+x)^m;</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left;text-indent:21.7=
5pt'><st1:place
w:st=3D"on"><st1:City w:st=3D"on"><span lang=3DEN-US>taylor</span></st1:Cit=
y></st1:place><span
lang=3DEN-US>(f,5)</span></p>
<p class=3DMsoNormal align=3Dleft style=3D'text-align:left'><span style=3D'=
font-family:
SimSun;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times=
New Roman"'>运行结果为:</span></p>
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