Editing Appendix/Ramblings/PowerSeriesExpressions (section)

==Taylor Series (Hunter77)==

===First (Unsuccessful) Try===

First:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (- 3\Delta) f_3^' + \frac{1}{2} (- 3\Delta)^2 f^{''}_3 + \frac{1}{6} (- 3\Delta)^3 f_3^{'''} + \frac{1}{24}(- 3\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 - (3\Delta) f_3^' + \frac{3^2}{2} (\Delta)^2 f^{''}_3 - \frac{3^2}{2} (\Delta)^3 f_3^{'''} + \frac{3^3}{2^3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
- \frac{3^2}{2} (\Delta)^2 f^{''}_3 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 - f_0 - (3\Delta) f_3^' - \frac{3^2}{2} (\Delta)^3 f_3^{'''} + \frac{3^3}{2^3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

Note that, replacing the <math>~(\Delta)^3 f_3^{'''}</math> term with the expression derived in the ''Second'' step, below, gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
- \frac{3^2}{2} (\Delta)^2 f^{''}_3 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 - f_0 - (3\Delta) f_3^' + \frac{3^3}{2^3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{3^2}{2} \biggl\{  
\biggl[\frac{2^2}{3^2} \biggr] f_0
- f_1
+ f_3 \biggl[\frac{5}{3^2} \biggr] 
+  \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^' 
+ \biggl[- \frac{5}{6} \biggr] (\Delta)^4 f_3^{iv} 
\biggr\}\biggl[ -\frac{3}{2} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; 
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 - f_0 - 3 (\Delta) f_3^' + \frac{3^3}{2^3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{  
3  f_0
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[\frac{15}{2^2} \biggr]  f_3 
+ \biggl[- \frac{3}{2}\biggr] (\Delta) f_3^' 
+ \biggl[- \frac{3^2\cdot 5}{2^3} \biggr] (\Delta)^4 f_3^{iv} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; 
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2f_0 
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[1 + \frac{15}{2^2} \biggr] f_3 +  \biggl[-3 - \frac{3}{2}\biggr] (\Delta) f_3^' + \biggl[\frac{3^3}{2^3}- \frac{3^2\cdot 5}{2^3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; 
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2f_0 
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[\frac{19}{2^2} \biggr] f_3 +  \biggl[- \frac{9}{2}\biggr] (\Delta) f_3^' + \biggl[- \frac{9}{4} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

Then, replacing the <math>~(\Delta)^4 f_3^{iv}</math> term with the expression derived in the ''Third'' step, below, gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
- \frac{3^2}{2} (\Delta)^2 f^{''}_3 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2f_0 
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[\frac{19}{2^2} \biggr] f_3 +  \biggl[- \frac{9}{2}\biggr] (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[- \frac{9}{4} \biggr] \biggl\{
\biggl[-\frac{1}{3^2}  \biggr]f_0 
+ \biggl[\frac{1}{2} \biggr] f_1
- f_2
+ \biggl[ \frac{11}{2\cdot 3^2} \biggr] f_3 
+  \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^' 
\biggr\}\biggl[- 2^2\cdot 3 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2f_0 
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[\frac{19}{2^2} \biggr] f_3 +  \biggl[- \frac{9}{2}\biggr] (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{
\biggl[-3  \biggr]f_0 
+ \biggl[\frac{3^3 }{2} \biggr] f_1
- 3^3 f_2
+ \biggl[ \frac{3  \cdot 11}{2} \biggr] f_3 
+  \biggl[- 2\cdot 3^2 \biggr] (\Delta) f_3^' 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- f_0 
+ \biggl[\frac{3^3 }{2} - \frac{3^3}{2^2}\biggr] f_1
- 3^3 f_2
+ \biggl[\frac{3  \cdot 11}{2} + \frac{19}{2^2} \biggr] f_3 +  \biggl[- 2\cdot 3^2- \frac{9}{2}\biggr] (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- f_0 
+ \biggl[\frac{3^3}{2^2}\biggr] f_1
- 3^3 f_2
+ \biggl[\frac{5\cdot 17}{2^2} \biggr] f_3 +  \biggl[- \frac{3^2\cdot 5}{2}\biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>


Second:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (- 2\Delta) f_3^' + \frac{1}{2} (- 2\Delta)^2 f^{''}_3 + \frac{1}{6} (- 2\Delta)^3 f_3^{'''} + \frac{1}{24}(- 2\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 - 2(\Delta) f_3^' + 2 (\Delta)^2 f^{''}_3 - \frac{2^2}{3} (\Delta)^3 f_3^{'''} + \frac{2}{3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 - 2(\Delta) f_3^' 
- \frac{2^2}{3} (\Delta)^3 f_3^{'''} + \frac{2}{3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[\frac{2^2}{3^2} \biggr] \biggl[ f_3 - f_0 - (3\Delta) f_3^' - \frac{3^2}{2} (\Delta)^3 f_3^{'''} + \frac{3^3}{2^3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2^2}{3^2} \biggr] f_0
+ f_3 \biggl[1-\frac{2^2}{3^2} \biggr] 
+ \biggl[ \frac{2^2}{3^2} (3\Delta) - 2(\Delta) \biggr] f_3^' 
+ \biggl[ \biggl(\frac{2^2}{3^2} \biggr) \frac{3^2}{2} (\Delta)^3  - \frac{2^2}{3} (\Delta)^3 \biggr] f_3^{'''} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ \frac{2}{3}(\Delta)^4 - \biggl( \frac{2^2}{3^2} \biggr) \frac{3^3}{2^3}(\Delta)^4 \biggr]  f_3^{iv} 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2^2}{3^2} \biggr] f_0
+ f_3 \biggl[\frac{5}{3^2} \biggr] 
+  \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^' 
+ \biggl[ \frac{2}{3} \biggr] (\Delta)^3f_3^{'''} 
+ \biggl[- \frac{5}{6} \biggr] (\Delta)^4 f_3^{iv} 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
- \biggl[ \frac{2}{3} \biggr] (\Delta)^3f_3^{'''} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2^2}{3^2} \biggr] f_0
- f_1
+ f_3 \biggl[\frac{5}{3^2} \biggr] 
+  \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^' 
+ \biggl[- \frac{5}{6} \biggr] (\Delta)^4 f_3^{iv} 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

Now, replacing the <math>~(\Delta)^4 f_3^{iv}</math> term with the expression derived in the ''Third'' step, below, gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
- \biggl[ \frac{2}{3} \biggr] (\Delta)^3f_3^{'''} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2^2}{3^2} \biggr] f_0
- f_1
+ f_3 \biggl[\frac{5}{3^2} \biggr] 
+  \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[- \frac{5}{6} \biggr] \biggl\{
\biggl[-\frac{1}{3^2}  \biggr]f_0 
+ \biggl[\frac{1}{2} \biggr] f_1
- f_2
+ \biggl[ \frac{11}{2\cdot 3^2} \biggr] f_3 
+  \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^' 
\biggr\} \biggl[ -2^2\cdot 3\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2^2}{3^2} \biggr] f_0
- f_1
+ f_3 \biggl[\frac{5}{3^2} \biggr] 
+  \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{
\biggl[-\frac{2\cdot 5 }{3^2}  \biggr]f_0 
+ \biggl[5\biggr] f_1
+ \biggl[- 2\cdot 5 \biggr] f_2
+ \biggl[ \frac{5\cdot 11}{3^2} \biggr] f_3 
+  \biggl[- \frac{2^2\cdot 5}{3}\biggr] (\Delta) f_3^' 
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2^2}{3^2} -\frac{2\cdot 5 }{3^2}  \biggr] f_0
+ \biggl[4\biggr] f_1
+ \biggl[- 2\cdot 5 \biggr] f_2
+ \biggl[\frac{5}{3^2} + \frac{5\cdot 11}{3^2}\biggr] f_3 
+  \biggl[- \frac{2^2\cdot 5}{3} - \frac{2}{3}\biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[-\frac{2}{3}  \biggr] f_0
+ \biggl[4\biggr] f_1
+ \biggl[- 2\cdot 5 \biggr] f_2
+ \biggl[\frac{2^2\cdot 5}{3}\biggr] f_3 
+  \biggl[- \frac{2\cdot 11}{3}\biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>


Third:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (- \Delta) f_3^' + \frac{1}{2} (- \Delta)^2 f^{''}_3 + \frac{1}{6} (- \Delta)^3 f_3^{'''} + \frac{1}{24}(- \Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + \biggl[ -1 \biggr](\Delta) f_3^' + \biggl[ \frac{1}{2} \biggr] (\Delta)^2 f^{''}_3 + \biggl[ - \frac{1}{2\cdot 3} \biggr] (\Delta)^3 f_3^{'''} + \biggl[ \frac{1}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + \biggl[ -1 \biggr](\Delta) f_3^' + \biggl[ \frac{1}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ \frac{1}{2} \biggr] \biggl\{ 
2f_0 
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[\frac{19}{2^2} \biggr] f_3 +  \biggl[- \frac{9}{2}\biggr] (\Delta) f_3^' + \biggl[- \frac{9}{4} \biggr] (\Delta)^4 f_3^{iv} 
\biggr\} \biggl[-\frac{2}{3^2}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ - \frac{1}{2\cdot 3} \biggr] \biggl\{ 
\biggl[\frac{2^2}{3^2} \biggr] f_0
- f_1
+ f_3 \biggl[\frac{5}{3^2} \biggr] 
+  \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^' 
+ \biggl[- \frac{5}{6} \biggr] (\Delta)^4 f_3^{iv} 
\biggr\} \biggl[-\frac{3}{2}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + \biggl[ -1 \biggr](\Delta) f_3^' + \biggl[ \frac{1}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{ 
\biggl[ -\frac{2}{3^2} \biggr]f_0 
+ \biggl[\frac{3}{2^2}\biggr] f_1
+ \biggl[-\frac{19}{2^2\cdot 3^2} \biggr] f_3 +  \biggl[\frac{1}{2}\biggr] (\Delta) f_3^' + \biggl[\frac{1}{4} \biggr] (\Delta)^4 f_3^{iv} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{ 
\biggl[\frac{1}{3^2} \biggr] f_0
+ \biggl[- \frac{1}{2^2} \biggr] f_1
+ f_3 \biggl[\frac{5}{2^2\cdot 3^2} \biggr] 
+  \biggl[- \frac{1}{2\cdot 3}\biggr] (\Delta) f_3^' 
+ \biggl[- \frac{5}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + \biggl[ -1 \biggr](\Delta) f_3^' + \biggl[ \frac{1}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{ 
\biggl[\frac{1}{3^2}  -\frac{2}{3^2} \biggr]f_0 
+ \biggl[\frac{3}{2^2}- \frac{1}{2^2} \biggr] f_1
+ \biggl[\frac{5}{2^2\cdot 3^2} -\frac{19}{2^2\cdot 3^2} \biggr] f_3 +  \biggl[\frac{1}{2}- \frac{1}{2\cdot 3}\biggr] (\Delta) f_3^' 
+ \biggl[\frac{1}{4} - \frac{5}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[-\frac{1}{3^2}  \biggr]f_0 
+ \biggl[\frac{1}{2} \biggr] f_1
+ \biggl[ \frac{11}{2\cdot 3^2} \biggr] f_3 
+  \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^' 
+ \biggl[\frac{1}{2^2\cdot 3} \biggr] (\Delta)^4 f_3^{iv} 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
- \biggl[\frac{1}{2^2\cdot 3} \biggr] (\Delta)^4 f_3^{iv} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[-\frac{1}{3^2}  \biggr]f_0 
+ \biggl[\frac{1}{2} \biggr] f_1
- f_2
+ \biggl[ \frac{11}{2\cdot 3^2} \biggr] f_3 
+  \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

And, finally:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (\Delta) f_3^' + \frac{1}{2} ( \Delta)^2 f^{''}_3 + \frac{1}{6} (\Delta)^3 f_3^{'''} + \frac{1}{24}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2} \biggl\{
- f_0 
+ \biggl[\frac{3^3}{2^2}\biggr] f_1
- 3^3 f_2
+ \biggl[\frac{5\cdot 17}{2^2} \biggr] f_3 +  \biggl[- \frac{3^2\cdot 5}{2}\biggr] (\Delta) f_3^' 
\biggr\} \biggl[ - \frac{2}{3^2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{6} \biggl\{
\biggl[-\frac{2}{3}  \biggr] f_0
+ \biggl[4\biggr] f_1
+ \biggl[- 2\cdot 5 \biggr] f_2
+ \biggl[\frac{2^2\cdot 5}{3}\biggr] f_3 
+  \biggl[- \frac{2\cdot 11}{3}\biggr] (\Delta) f_3^' 
\biggr\} \biggl[ -\frac{3}{2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{24}\biggl\{
\biggl[-\frac{1}{3^2}  \biggr]f_0 
+ \biggl[\frac{1}{2} \biggr] f_1
- f_2
+ \biggl[ \frac{11}{2\cdot 3^2} \biggr] f_3 
+  \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^' 
\biggr\} \biggl[ -2^2\cdot 3 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{
\biggl[ \frac{1}{3^2}  \biggr] f_0 
+ \biggl[- \frac{3}{2^2}\biggr] f_1
+\biggl[ 3 \biggr] f_2
+ \biggl[- \frac{5\cdot 17}{2^2\cdot 3^2} \biggr] f_3 
+  \biggl[\frac{5}{2}\biggr] (\Delta) f_3^' 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{2\cdot 3}  \biggr] f_0
+ \biggl[-1 \biggr] f_1
+ \biggl[ \frac{5}{2} \biggr] f_2
+ \biggl[- \frac{5}{3}\biggr] f_3 
+  \biggl[\frac{11}{2\cdot 3}\biggr] (\Delta) f_3^' 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{2\cdot 3^2}  \biggr]f_0 
+ \biggl[- \frac{1}{2^2} \biggr] f_1
+ \biggl[ \frac{1}{2} \biggr] f_2
+ \biggl[ -\frac{11}{2^2 \cdot 3^2} \biggr] f_3 
+  \biggl[\frac{1}{3}\biggr] (\Delta) f_3^' 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{1}{3^2} + \frac{1}{2\cdot 3} + \frac{1}{2\cdot 3^2}  \biggr] f_0 + 
\biggl[- \frac{3}{2^2} - 1 - \frac{1}{2^2} \biggr] f_1
+\biggl[ 3 + \frac{5}{2} + \frac{1}{2} \biggr] f_2
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[1 - \frac{5\cdot 17}{2^2\cdot 3^2} - \frac{5}{3} - \frac{11}{2^2 \cdot 3^2} \biggr] f_3 
+  \biggl[1 + \frac{5}{2} + \frac{11}{2\cdot 3} + \frac{1}{3} \biggr] (\Delta) f_3^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{1}{3} \biggr] f_0 + 
\biggl[- 2\biggr] f_1
+\biggl[ 6 \biggr] f_2
+ \biggl[- \frac{10}{3} \biggr] f_3 
+  \biggl[\frac{17}{3} \biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

Result:

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">
Definitely WRONG!
  </th>
</tr>
<tr><td>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{3} ~ f_0 - 2 f_1 + 6 f_2
- \frac{10}{3} ~ f_3 
+  \frac{17}{3}  (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5) \, .
</math>
  </td>
</tr>
</table>
</td></tr>
</table>
</div>

When I used an Excel spreadsheet to test this out against a parabola, the integration quickly became wildly unstable, strongly suggesting that there is an error in the derivation.  My first attempt to uncover this error produced a new coefficient on the <math>~(\Delta) f_3^'</math>, namely,

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">
Somewhat Improved
  </th>
</tr>
<tr><td>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{3} ~ f_0 - 2 f_1 + 6 f_2
- \frac{10}{3} ~ f_3 
+  4  (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5) \, .
</math>
  </td>
</tr>
</table>
</td></tr>
</table>
</div>
Although it showed improvement, this expression still blows up.  So I have not bothered to revise the original (definitely WRONG!) derivation.  Instead, let's start all over and approach it with a more gradual derivation.

===Second Try===

We will work from the following foundation expression in which <math>~f_4</math> is the variable that we desire to evaluate, and the "known" quantities are: &nbsp; <math>~f_3</math>, <math>~f_3^'</math>,  <math>~f_2</math>, <math>~f_1</math>, and <math>~f_0</math>.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (\Delta) f_3^' + \frac{1}{2} ( \Delta)^2 f^{''}_3 + \frac{1}{6} (\Delta)^3 f_3^{'''} + \frac{1}{24}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

Let's use similar Taylor-series expansions for <math>~f_2</math>, <math>~f_3</math>, etc. in order to eliminate the <math>~f_3^{''}</math> term,  the <math>~f_3^{'''}</math> term, etc.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (- \Delta) f_3^' + \frac{1}{2} (- \Delta)^2 f^{''}_3 + \frac{1}{6} (- \Delta)^3 f_3^{'''} + \frac{1}{24}(- \Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~f_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (- 2\Delta) f_3^' + \frac{1}{2} (- 2\Delta)^2 f^{''}_3 + \frac{1}{6} (- 2\Delta)^3 f_3^{'''} + \frac{1}{24}(- 2\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~f_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (- 3\Delta) f_3^' + \frac{1}{2} (- 3\Delta)^2 f^{''}_3 + \frac{1}{6} (- 3\Delta)^3 f_3^{'''} + \frac{1}{24}(- 3\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

---- 

First:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-\frac{1}{2} (- \Delta)^2 f^{''}_3 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (- \Delta) f_3^' - f_2+ \frac{1}{6} (- \Delta)^3 f_3^{'''} + \frac{1}{24}(- \Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\frac{1}{2} (\Delta)^2 f^{''}_3 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- f_3 + (\Delta) f_3^' + f_2+ \frac{1}{6} (\Delta)^3 f_3^{'''} - \frac{1}{24}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
f_4
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (\Delta) f_3^' - f_3 + (\Delta) f_3^' + f_2 + \mathcal{O}(\Delta^3)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_2 + 2(\Delta) f_3^' + \mathcal{O}(\Delta^3)
</math>
  </td>
</tr>
</table>
</div>

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr><td align="center"><math>~\mathcal{O}(\Delta^3)</math></td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
f_4
</math>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_2 + 2(\Delta) f_3^' + \mathcal{O}(\Delta^3)
</math>
  </td>
</tr>
</table>
</td></tr></table>
</div>

This expression works very well for a parabola.

----

Second:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (- 2) \Delta f_3^' + 2 (\Delta)^2 f^{''}_3 + \biggl[- \frac{2^3}{6}\biggr]  \Delta^3 f_3^{'''} + \biggl[ \frac{2^4}{2^3\cdot 3} \biggr] \Delta^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (- 2) \Delta f_3^' + \biggl[- \frac{2^3}{6}\biggr]  \Delta^3 f_3^{'''} + \biggl[ \frac{2^4}{2^3\cdot 3} \biggr] \Delta^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2 \biggl\{
- f_3 + (\Delta) f_3^' + f_2+ \frac{1}{2\cdot 3} (\Delta)^3 f_3^{'''} - \frac{1}{2^3\cdot 3}(\Delta)^4 f_3^{iv}  
\biggr\} \biggl[ 2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3\biggl[ 1 - 2^2\biggr] + (2^2 - 2) \Delta f_3^' + 2^2f_2
+  \biggl[\frac{2}{3} - \frac{2^3}{6}\biggr]  \Delta^3 f_3^{'''} 
+ \biggl[ \frac{2^4}{2^3\cdot 3} - \frac{1}{2\cdot 3}\biggr] \Delta^4 f_3^{iv} 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3\biggl[ -3\biggr] + (2) \Delta f_3^' + 2^2f_2
+  \biggl[- \frac{2}{3}\biggr]  \Delta^3 f_3^{'''} 
+ \biggl[ \frac{1}{2} \biggr] \Delta^4 f_3^{iv} 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\biggl[\frac{2}{3}\biggr]  \Delta^3 f_3^{'''} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^' 
+ \biggl[ \frac{1}{2} \biggr] \Delta^4 f_3^{iv} 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

This also allows us to improve the expression for the <math>~f_3^{''}</math> term, as initially derived in the "First" subsection, above.  Namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{1}{2} (\Delta)^2 f^{''}_3 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_2 - f_3 + (\Delta) f_3^' - \frac{1}{24}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{6} \biggl\{
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^' 
+ \biggl[ \frac{1}{2} \biggr] \Delta^4 f_3^{iv} 
\biggr\} \biggl[ \frac{3}{2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{4} f_1 + 2f_2 + \biggl[ - \frac{7}{4} \biggr] f_3 + \biggl[ \frac{3}{2} \biggr] (\Delta) f_3^' 
+ \biggl[\frac{1}{2^2\cdot 3} \biggr](\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

Hence, an improved expression for <math>~f_4</math> is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^4)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{
- \frac{1}{4} f_1 + 2f_2 + \biggl[ - \frac{7}{4} \biggr] f_3 + \biggl[ \frac{3}{2} \biggr] (\Delta) f_3^' 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{6} \biggl\{
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^' 
\biggr\} \biggl[ \frac{3}{2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{2} f_1 + 3f_2 - \frac{3}{2} f_3 + 3(\Delta) f_3^' + \mathcal{O}(\Delta^4)
</math>
  </td>
</tr>
</table>
</div>


<div align="center">
<table border="1" cellpadding="8" align="center">
<tr><td align="center"><math>~\mathcal{O}(\Delta^4)</math></td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
f_4
</math>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{2} f_1 + 3f_2 - \frac{3}{2} f_3 + 3(\Delta) f_3^' + \mathcal{O}(\Delta^4)
</math>
  </td>
</tr>
</table>
</td></tr></table>
</div>


----

Third:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (- 3\Delta) f_3^' + \frac{1}{2} (- 3\Delta)^2 f^{''}_3 + \frac{1}{6} (- 3\Delta)^3 f_3^{'''} + \frac{1}{24}(- 3\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + \biggl[ - 3 \biggr] (\Delta) f_3^'  + \biggl[ \frac{3^3}{2^3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 3^2 \biggl\{
- \frac{1}{4} f_1 + 2f_2 + \biggl[ - \frac{7}{4} \biggr] f_3 + \biggl[ \frac{3}{2} \biggr] (\Delta) f_3^' 
+ \biggl[\frac{1}{2^2\cdot 3} \biggr](\Delta)^4 f_3^{iv} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[-\frac{3^3}{2^2} \biggr] \biggl\{
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^' 
+ \biggl[ \frac{1}{2} \biggr] \Delta^4 f_3^{iv} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + \biggl[ - 3 \biggr] (\Delta) f_3^'  + \biggl[ \frac{3^3}{2^3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{
\biggl[- \frac{3^2 }{4} \biggr]  f_1 + \biggl[ 2\cdot 3^2 \biggr]f_2 + \biggl[ - \frac{3^2 \cdot 7}{4} \biggr] f_3 + \biggl[ \frac{3^3}{2} \biggr] (\Delta) f_3^' 
+ \biggl[\frac{3}{2^2} \biggr](\Delta)^4 f_3^{iv} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \biggl\{
\biggl[\frac{3^3}{2^2} \biggr] f_1 
+ \biggl[-3^3 \biggr] f_2 
+ \biggl[\frac{3^4}{2^2} \biggr]f_3 
+ \biggl[- \frac{3^3}{2} \biggr] \Delta f_3^' 
+ \biggl[- \frac{3^3}{2^3} \biggr] \Delta^4 f_3^{iv} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{3^3}{2^2} - \frac{3^2 }{4} \biggr]  f_1 + \biggl[ 2\cdot 3^2 -3^3\biggr]f_2 + \biggl[ 1+ \frac{3^4}{2^2} - \frac{3^2 \cdot 7}{4} \biggr] f_3 
+ \biggl[ \frac{3^3}{2} - \frac{3^3}{2} -3\biggr] (\Delta) f_3^' 
+ \biggl[\frac{3^3}{2^3}  + \frac{3}{2^2} - \frac{3^3}{2^3}\biggr](\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{3^2}{2} \biggr]  f_1 + \biggl[ - 3^2 \biggr]f_2 + \biggl[ \frac{11}{2} \biggr] f_3 
+ \biggl[ -3\biggr] (\Delta) f_3^' 
+ \biggl[\frac{3}{2^2} \biggr](\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
- \biggl[\frac{3}{2^2} \biggr](\Delta)^4 f_3^{iv} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- f_0 + \biggl[\frac{3^2}{2} \biggr]  f_1 + \biggl[ - 3^2 \biggr]f_2 + \biggl[ \frac{11}{2} \biggr] f_3 
+ \biggl[ -3\biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{1}{2} (\Delta)^2 f^{''}_3 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{4} f_1 + 2f_2 + \biggl[ - \frac{7}{4} \biggr] f_3 + \biggl[ \frac{3}{2} \biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{1}{2^2\cdot 3} \biggr]\biggl\{
- f_0 + \biggl[\frac{3^2}{2}\biggr]  f_1 + \biggl[ - 3^2 \biggr]f_2 + \biggl[ \frac{11}{2} \biggr] f_3 
+ \biggl[ -3\biggr] (\Delta) f_3^' 
\biggr\} \biggl[ - \frac{2^2}{3}   \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{4} f_1 + 2f_2 + \biggl[ - \frac{7}{4} \biggr] f_3 + \biggl[ \frac{3}{2} \biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{1}{2} \biggr] f_1 
+ f_2 + \biggl[- \frac{11}{2 \cdot 3^2} \biggr] f_3 
+ \biggl[\frac{1}{3} \biggr] (\Delta) f_3^' 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{1}{2} - \frac{1}{4} \biggr] f_1 
+ 3 f_2 + \biggl[- \frac{11}{2 \cdot 3^2} - \frac{7}{4} \biggr] f_3 
+ \biggl[\frac{1}{3} +  \frac{3}{2} \biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{3}{4} \biggr] f_1 
+ 3 f_2 + \biggl[- \frac{5\cdot 17}{2^2\cdot 3^2} \biggr] f_3 
+ \biggl[\frac{11}{2\cdot 3} \biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

And,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl[\frac{2}{3}\biggr]  \Delta^3 f_3^{'''} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ \frac{1}{2} \biggr] \biggl\{
- f_0 + \biggl[\frac{3^2}{2} \biggr]  f_1 + \biggl[ - 3^2 \biggr]f_2 + \biggl[ \frac{11}{2} \biggr] f_3 
+ \biggl[ -3\biggr] (\Delta) f_3^' 
\biggr\} \biggl[ - \frac{2^2}{3}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{
\biggl[ \frac{2}{3}  \biggr] f_0 + \biggl[- 3 \biggr]  f_1 
+ \biggl[ 2\cdot 3 \biggr] f_2 + \biggl[- \frac{11}{3}  \biggr] f_3 
+ \biggl[ 2 \biggr] (\Delta) f_3^' 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2}{3}  \biggr] f_0 + \biggl[- 4 \biggr]  f_1 
+ \biggl[ 2\cdot 5 \biggr] f_2 + \biggl[- \frac{2^2 \cdot 5}{3}  \biggr] f_3 
+ \biggl[ 4 \biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>

----

Finally, then:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (\Delta) f_3^' + \frac{1}{2} ( \Delta)^2 f^{''}_3 + \frac{1}{6} (\Delta)^3 f_3^{'''} + \frac{1}{24}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2} \biggl\{
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{3}{4} \biggr] f_1 
+ 3 f_2 + \biggl[- \frac{5\cdot 17}{2^2\cdot 3^2} \biggr] f_3 
+ \biggl[\frac{11}{2\cdot 3} \biggr] (\Delta) f_3^' 
\biggr\}\biggl[ 2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \frac{1}{2\cdot 3} \biggl\{
\biggl[ \frac{2}{3}  \biggr] f_0 + \biggl[- 4 \biggr]  f_1 
+ \biggl[ 2\cdot 5 \biggr] f_2 + \biggl[- \frac{2^2 \cdot 5}{3}  \biggr] f_3 
+ \biggl[ 4 \biggr] (\Delta) f_3^' 
\biggr\}\biggl[ \frac{3}{2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^3\cdot 3} \biggl\{
- f_0 + \biggl[\frac{3^2}{2} \biggr]  f_1 + \biggl[ - 3^2 \biggr]f_2 + \biggl[ \frac{11}{2} \biggr] f_3 
+ \biggl[ -3\biggr] (\Delta) f_3^' 
\biggr\}\biggl[- \frac{2^2}{3}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{3}{4} \biggr] f_1 
+ 3 f_2 + \biggl[- \frac{5\cdot 17}{2^2\cdot 3^2} \biggr] f_3 
+ \biggl[\frac{11}{2\cdot 3} \biggr] (\Delta) f_3^' 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \frac{1}{2^2} \biggl\{
\biggl[ \frac{2}{3}  \biggr] f_0 + \biggl[- 4 \biggr]  f_1 
+ \biggl[ 2\cdot 5 \biggr] f_2 + \biggl[- \frac{2^2 \cdot 5}{3}  \biggr] f_3 
+ \biggl[ 4 \biggr] (\Delta) f_3^' 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2\cdot 3^2} \biggl\{
f_0 + \biggl[ - \frac{3^2}{2} \biggr]  f_1 + \biggl[ 3^2 \biggr]f_2 + \biggl[- \frac{11}{2} \biggr] f_3 
+ \biggl[ 3\biggr] (\Delta) f_3^' 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{3}{4} \biggr] f_1 
+ 3 f_2 + \biggl[- \frac{5\cdot 17}{2^2\cdot 3^2} \biggr] f_3 
+ \biggl[\frac{11}{2\cdot 3} \biggr] (\Delta) f_3^' 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \biggl\{
\biggl[ \frac{1}{2\cdot 3}  \biggr] f_0 + \biggl[- 1 \biggr]  f_1 
+ \biggl[ \frac{5}{2} \biggr] f_2 + \biggl[- \frac{5}{3}  \biggr] f_3 
+ \biggl[ 1 \biggr] (\Delta) f_3^' 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\biggl\{
\biggl[ \frac{1}{2\cdot 3^2} \biggr] f_0 + \biggl[ - \frac{1}{2^2} \biggr]  f_1 + \biggl[ \frac{1}{2} \biggr]f_2 + \biggl[- \frac{11}{2^2\cdot 3^2} \biggr] f_3 
+ \biggl[ \frac{1}{2\cdot 3} \biggr] (\Delta) f_3^' 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{1}{3^2} + \frac{1}{2\cdot 3} + \frac{1}{2\cdot 3^2} \biggr] f_0 
+ \biggl[- 1  - \frac{3}{4} - \frac{1}{2^2} \biggr] f_1 
+ \biggl[ 3 + \frac{5}{2} + \frac{1}{2} \biggr] f_2 
+ \biggl[1 - \frac{5\cdot 17}{2^2\cdot 3^2} - \frac{5}{3} - \frac{11}{2^2\cdot 3^2} \biggr] f_3 
+ \biggl[2 + \frac{11}{2\cdot 3} + \frac{1}{2\cdot 3} \biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{1}{3}  \biggr] f_0 
+ \biggl[- 2\biggr] f_1 
+ \biggl[ 6 \biggr] f_2 
+ \biggl[- \frac{2\cdot 5}{3}  \biggr] f_3 
+ \biggl[4 \biggr] (\Delta) f_3^' 
+ \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</div>


<div align="center">
<table border="1" cellpadding="8" align="center">
<tr><td align="center"><math>~\mathcal{O}(\Delta^5)</math></td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
f_4
</math>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{3}  f_0 - 2 f_1 + 6 f_2 - \frac{2\cdot 5}{3} f_3 + 4  (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
  </td>
</tr>
</table>
</td></tr></table>
</div>



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