Originally Posted by

**Deadstar** Perhaps this is the wrong method but I'll post it up...

For the case when $\displaystyle f$ is even, $\displaystyle f(x) = f(-x)$.

$\displaystyle f(x) = a_0 + a_1 x + a_2x^2 + a_3 x^3 + \dots$

$\displaystyle f(-x) = a_0 - a_1 x + a_2 x^2 - a_3 x^3 + \dots$

So equating the series you get...

$\displaystyle a_0 + a_1 x + a_2x^2 + a_3 x^3 + \dots = a_0 - a_1 x + a_2x^2 - a_3 x^3 + \dots$

Canceling gives you...

$\displaystyle a_1 x + a_3 x^3 + \dots = - a_1 x - a_3 x^3 - \dots$

Hence equating coefficients gives...

$\displaystyle a_1 = -a_1$, $\displaystyle a_1 = 0$,

$\displaystyle a_3 = -a_3$, $\displaystyle a_3 = 0$,

etc...

Similar for the $\displaystyle f$ is odd case (in which $\displaystyle f(x) = -f(-x)$)