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Notebook[{
Cell[TextData["Plotting Partial Sums of Fourier Series"], "Title",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["Needs[\"Graphics`Colors`\"]", "Input",
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Cell[TextData[{
  "We will construct partial sums of the Fourier series for ",
  " ",
  Cell[BoxData[
      \(TraditionalForm\`x\^2\)]],
  " on [-2,2]."
}], "Text"],

Cell[BoxData[{
    \(\(L = 2;\)\), "\[IndentingNewLine]", 
    \(\(f[x_] = x\^2;\)\)}], "Input"],

Cell["The Fourier coefficients are", "Text"],

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Cell[TextData[{
  "Here is the ",
  Cell[BoxData[
      \(TraditionalForm\`k\^th\)]],
  " partial sum of the Fourier series for f[x]."
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(fourierSum[x_, k_]\  := \ 
      a[0] + \[Sum]\+\(n = 1\)\%k\((a[n]\ Cos[\(n\ \[Pi]\ x\)\/L] + 
              b[n]\ Sin[\(n\ \[Pi]\ x\)\/L])\)\)], "Input"],

Cell[TextData[{
  "These are plots of the ",
  Cell[BoxData[
      \(TraditionalForm\`0\^th\)]],
  " through the ",
  Cell[BoxData[
      \(TraditionalForm\`5\^th\)]],
  " partial sums.  Click on any one of the graphs to view an animation."
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
Table[
  Plot[{f[x],fourierSum[x,k]}, {x,-2L,2L},
      PlotStyle->{{Black},{Red}},
      PlotRange->{-1,6}],
  {k,0,5}
];\
\>", "Input",
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