Add factorial in denominator of cancelling cubic terms.

porcuquine · porcuquine · commit b577200d1fca · 2017-02-19T16:46:16.000-08:00
diff --git a/lessons/04_Step_3.ipynb b/lessons/04_Step_3.ipynb
@@ -63,9 +63,9 @@
    "source": [
     "The second-order derivative can be represented geometrically as the line tangent to the curve given by the first derivative.  We will discretize the second-order derivative with a Central Difference scheme: a combination of Forward Difference and Backward Difference of the first derivative.  Consider the Taylor expansion of $u_{i+1}$ and $u_{i-1}$ around $u_i$:\n",
     "\n",
-    "$u_{i+1} = u_i + \\Delta x \\frac{\\partial u}{\\partial x}\\bigg|_i + \\frac{\\Delta x^2}{2} \\frac{\\partial ^2 u}{\\partial x^2}\\bigg|_i + \\frac{\\Delta x^3}{3} \\frac{\\partial ^3 u}{\\partial x^3}\\bigg|_i + O(\\Delta x^4)$\n",
+    "$u_{i+1} = u_i + \\Delta x \\frac{\\partial u}{\\partial x}\\bigg|_i + \\frac{\\Delta x^2}{2} \\frac{\\partial ^2 u}{\\partial x^2}\\bigg|_i + \\frac{\\Delta x^3}{3!} \\frac{\\partial ^3 u}{\\partial x^3}\\bigg|_i + O(\\Delta x^4)$\n",
     "\n",
-    "$u_{i-1} = u_i - \\Delta x \\frac{\\partial u}{\\partial x}\\bigg|_i + \\frac{\\Delta x^2}{2} \\frac{\\partial ^2 u}{\\partial x^2}\\bigg|_i - \\frac{\\Delta x^3}{3} \\frac{\\partial ^3 u}{\\partial x^3}\\bigg|_i + O(\\Delta x^4)$\n",
+    "$u_{i-1} = u_i - \\Delta x \\frac{\\partial u}{\\partial x}\\bigg|_i + \\frac{\\Delta x^2}{2} \\frac{\\partial ^2 u}{\\partial x^2}\\bigg|_i - \\frac{\\Delta x^3}{3!} \\frac{\\partial ^3 u}{\\partial x^3}\\bigg|_i + O(\\Delta x^4)$\n",
     "\n",
     "If we add these two expansions, you can see that the odd-numbered derivative terms will cancel each other out.  If we neglect any terms of $O(\\Delta x^4)$ or higher (and really, those are very small), then we can rearrange the sum of these two expansions to solve for our second-derivative.  \n"
    ]
