Another Look at Series
Another Look at Series
In[31]:=
Series[Sin[x],{x,0,10}]
Out[31]=
3 5 7 9
x x x x 11
x - -- + --- - ---- + ------ + O[x]
6 120 5040 362880
In[32]:=
Series[Cos[x],{x,0,10}]
Out[32]=
2 4 6 8 10
x x x x x 11
1 - -- + -- - --- + ----- - ------- + O[x]
2 24 720 40320 3628800
Did you notice even and odd functions at work?
In[33]:=
Normal[Series[Exp[x],{x,0,10}]]
Out[33]=
2 3 4 5 6 7 8 9
x x x x x x x x
1 + x + -- + -- + -- + --- + --- + ---- + ----- + ------ +
2 6 24 120 720 5040 40320 362880
10
x
-------
3628800
The Normal command truncates the series so you can work with it.
In[34]:=
evenpart[Exp,x]
oddpart[Exp,x]
Out[34]=
-x x
E + E
--------
2
Out[35]=
-x x
-E + E
---------
2
In[36]:=
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];
In[37]:=
Series[Exp[x],{x,0,10}]
Series[evenpart[Exp,x],{x,0,10}]
Series[oddpart[Exp,x],{x,0,10}]
Out[37]=
2 3 4 5 6 7 8 9
x x x x x x x x
1 + x + -- + -- + -- + --- + --- + ---- + ----- + ------ +
2 6 24 120 720 5040 40320 362880
10
x 11
------- + O[x]
3628800
Out[38]=
2 4 6 8 10
x x x x x 11
1 + -- + -- + --- + ----- + ------- + O[x]
2 24 720 40320 3628800
Out[39]=
3 5 7 9
x x x x 11
x + -- + --- + ---- + ------ + O[x]
6 120 5040 362880
Now did you notice something?
Keep this in mind when we do orthogonal expansions!
Up to Even and Odd Functions