Another Example

Let's do an one more example. Use the odd periodic extension (hence the sine series) for this function. This function should look familiar.
f(x) = x for 0<=x<=1
2-x for1<=x<=2

In[71]:=

  Clear[p1,p2,fgraph]
  p1=Plot[x,{x,0,1},DisplayFunction->Identity];
  p2=Plot[2-x,{x,1,2},DisplayFunction->Identity];
  fgraph=Show[p1,p2,DisplayFunction->$DisplayFunction,
  AspectRatio->Automatic,
  Ticks->{{1,2},Automatic}];

The period here is 4, so L is 2 (after we do the periodic extension!).

In[72]:=

  b[n_]:=(2/2)(Integrate[x v[x,2,n],{x,0,1}]+
  Integrate[(2-x) v[x,2,n],{x,1,2}]);
  Table[b[k],{k,1,10}]

Out[72]=

    8       -8         8          -8         8
  {---, 0, -----, 0, ------, 0, ------, 0, ------, 0}
     2         2          2          2          2
   Pi      9 Pi      25 Pi      49 Pi      81 Pi

Can you see the pattern?

Can you see the pattern? ...

Let's calculate some partial sums. Be patient.

In[73]:=

  Clear[Sn]
  Sn=Table[Sum[b[k] v[x,2,k],{k,1,n}],{n,1,5}];

Let's see how we did.

In[74]:=

  Clear[sn1,sn3,sn5,sn10]

In[75]:=

  sn1=Plot[Sn[[1]],{x,0,2},DisplayFunction->Identity];
  Show[fgraph,sn1,PlotLabel->
  FontForm["b1sin[Pi/2 x]",{"Helvetica-Bold",12}]];

In[76]:=

  sn3=Plot[Sn[[3]],{x,0,2},DisplayFunction->Identity];
  Show[fgraph,sn3,PlotLabel->
  FontForm["b1sin[Pi/2 x]+b2sin[Pi x]+b3sin[3Pi/2 x]",{"Helvetica-Bold",12}]];

In[77]:=

  sn5=Plot[Sn[[5]],{x,0,2},DisplayFunction->Identity];
  Show[fgraph,sn5,PlotLabel->
  FontForm["Sum(k=1 to 5) bkSin[kPi/2 x]",{"Helvetica-Bold",12}]];

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