Another Look at Series

Let's look at the Taylor series for some familiar functions.


  Series[Sin[x],{x,0,10}]


  Series[Cos[x],{x,0,10}]

Did you notice even and odd functions at work?


  Normal[Series[Exp[x],{x,0,10}]]

The Normal command truncates the series so you can work with it.


  evenpart[Exp,x]
  oddpart[Exp,x]


  p=Plot[Exp[x],{x,-3,3},DisplayFunction->Identity];
  pe=Plot[evenpart[Exp,x],{x,-3,3},DisplayFunction->Identity];
  po=Plot[oddpart[Exp,x],{x,-3,3},DisplayFunction->Identity];
  Show[p,pe,po,DisplayFunction->$DisplayFunction];


  Series[Exp[x],{x,0,10}]
  Series[evenpart[Exp,x],{x,0,10}]
  Series[oddpart[Exp,x],{x,0,10}]

Now did you notice something?
Keep this in mind when we do orthogonal expansions!

Up to Even and Odd Functions