Suppose that $f(x)=\sum_{n=0}^{\infty }c_n{x^{n}}$ for all $x$

Question

Suppose that $f(x)=\sum_{n=0}^{\infty }c_n{x^{n}}$ for all $x.$

Show that if $f$ is an odd function, $c_{0}=c_{2}=\dots=0$

and if $f$ is an even function, $c_{1}=c_{3}=\dots=0$

Answer 1

$F$ is said to be even if $F(x) =F(-x)$ and odd if $F(x)=-F(-x)$.

Now use these to get your answer

Answer 2

Apart from using the definition of even and odd functions you have to know that if a power series vanishes then all the coefficients vanish. This requires repeated differentiation of the series. Any book on Complex Analysis will have a proof of the fact that a power series can be differentiated term by term any number of times.