Editing Appendix/Ramblings/PowerSeriesExpressions (section)

====Theta-Function Examples====

<font color="darkgreen"><b>When <math>n=0</math></b></font>, all of the terms in the power-series expression higher than quadratic go to zero.  As a result, we see that 
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<math>\theta_{n=0}</math>
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<math>=</math>
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<math>1 - \frac{\xi^2}{6} \, .</math>
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<font color="darkgreen"><b>When <math>n=1</math></b></font>, the first few terms in the power-series expression are,
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<math>\theta_{n=1}</math>
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<math>=</math>
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<math>
1 - \frac{\xi^2}{6} + \frac{\xi^4}{120}  - \frac{\xi^6}{5040} 
+ \frac{\xi^8}{362880}  + \cdots
</math>
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&nbsp;
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<math>=</math>
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<math>
1 - \frac{\xi^2}{3!} + \frac{\xi^4}{5!}  - \frac{\xi^6}{7!} 
+ \frac{\xi^8}{9!}  + \cdots
</math>
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which is consistent with the power-series expression for <math>(\sin \xi)/\xi</math>.

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<font color="darkgreen"><b>When <math>n=5</math></b></font>, the first few terms in the power-series expression are,
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<math>~\theta_{n=5}</math>
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<math>~=</math>
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<math>~
1 - \frac{\xi^2}{3!} + \biggl(\frac{n}{5!}\biggr) \xi^4 - \frac{n}{7!} \biggl( \frac{8n-5}{3} \biggr) \xi^6 
+ \frac{n}{9!}\biggl[ \frac{(122n^2 -183n + 70)}{9} \biggr] \xi^8 + \cdots
</math>
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&nbsp;
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<math>~=</math>
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<math>~
1 - \frac{\xi^2}{3!} + \frac{\xi^4}{4!}  - \frac{\xi^6}{6!} \biggl( \frac{5^2}{3} \biggr) 
+ \frac{\xi^8}{8!}\biggl[ \frac{5^2}{3} \cdot \frac{7^2 }{3} \biggr]  + \cdots
</math>
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This should be compared with the [[SSC/Structure/Polytropes/Analytic#Primary_E-Type_Solution_2|known analytic solution]], which is
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<math>(\Theta_H)_{n=5}</math>
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<math>=</math>
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<math>\biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-1/2} \, .</math>
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From the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]], we start with,
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<math>~(1+b)^{m}</math>
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<math>~=</math>
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<math>~
1+ mb + \biggl[ \frac{m(m-1)}{2!}\biggr] b^{2}  + \biggl[ \frac{m(m-1)(m-2)}{3!} \biggr] b^{3} + \biggl[ \frac{m(m-1)(m-2)(m-3)}{4!} \biggr] b^{4} + \dots
</math>
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then set <math>b=\xi^2/3</math> and <math>m = -1/2</math> to obtain:

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<math>\theta_{n=5}</math>
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<math>~=</math>
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<math>~
1+ m\biggl( \frac{\xi^2}{3}\biggr) 
+ \frac{1}{2!}\biggl[ m\biggl(m-1 \biggr)\biggr] \biggl( \frac{\xi^2}{3}\biggr)^{2}  
+ \frac{1}{3!}\biggl[ m \biggl(m-1 \biggr)\biggl(m-2\biggr) \biggr] \biggl( \frac{\xi^2}{3}\biggr)^{3} 
+ \frac{1}{4!}\biggl[ m \biggl(m-1\biggr)\biggl(m-2\biggr)\biggl(m-3\biggr) \biggr] \biggl( \frac{\xi^2}{3}\biggr)^{4} + \dots
</math>
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&nbsp;
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<math>~=</math>
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<math>~
1+ \biggl(-\frac{1}{2}\biggr)\biggl( \frac{\xi^2}{3}\biggr) 
+ \frac{1}{2!}\biggl[ -\frac{1}{2}\biggl(-\frac{1}{2}-1 \biggr)\biggr] \biggl( \frac{\xi^4}{3^2}\biggr)  
+ \frac{1}{3!}\biggl[ -\frac{1}{2} \biggl(-\frac{1}{2}-1 \biggr)\biggl(-\frac{1}{2}-2\biggr) \biggr] \biggl( \frac{\xi^6}{3^3}\biggr) 
+ \frac{1}{4!}\biggl[ -\frac{1}{2} \biggl(-\frac{1}{2}-1\biggr)\biggl(-\frac{1}{2}-2\biggr)\biggl(-\frac{1}{2}-3\biggr) \biggr] 
\biggl( \frac{\xi^8}{3^4}\biggr) + \dots
</math>
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&nbsp;
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<math>~=</math>
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<math>~
1 - \biggl( \frac{\xi^2}{2\cdot 3}\biggr) 
+ \frac{1}{2!}\biggl[ \frac{3}{4}\biggr] \biggl( \frac{\xi^4}{3^2}\biggr)  
+ \frac{1}{3!}\biggl[ -\frac{15}{8} \biggr] \biggl( \frac{\xi^6}{3^3}\biggr) 
+ \frac{1}{4!}\biggl[ \biggl(\frac{105}{16}\biggr) \biggr] 
\biggl( \frac{\xi^8}{3^4}\biggr) + \dots
</math>
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&nbsp;
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<math>~=</math>
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<math>~
1 - \frac{\xi^2}{3!} + \frac{\xi^4}{4!}  
- \frac{\xi^6}{6!}\biggl( \frac{5^2}{3} \biggr)  
+ \frac{\xi^8}{8!}\biggl(\frac{5^2 \cdot 7^2}{3^2}\biggr) + \dots
</math>
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<b>QED</b>