Perhaps this is the wrong method but I'll post it up...
For the case when is even, .
So equating the series you get...
Canceling gives you...
Hence equating coefficients gives...
, ,
, ,
etc...
Similar for the is odd case (in which )
I do not understand how to go about the following proof, any suggestions?
Suppose f(x) = a0 + a1x + a2x2 + a3x3 + a4x4 + ... for x in an interval
(-R, R).
If f is even, prove that:
0 = a1 = a3 = a5 = a7 = ...
If f is odd, prove that:
0 = a0 = a2 = a4 = a6 = ...
Hint. The coefficients of a power series converging on an interval have to
be the Taylor coefficients of the function that the series converges to. Thus
it is not possible for different power series to converge to the same function
on an interval (-R, R).