Taylor serie for even function. Proof

Question

We let $f:\mathbb{R}\to\mathbb{R}$ be infinitely often differentiable function and we let the Taylor series be: $$\displaystyle\sum_{n=0}^{\infty}\left(\left(\frac{f^{n}(0)}{n!}\right)x^n\right) $$Let $\{a_n\}_{n\in \mathbb N }$ be $a_n=\frac{f^{n}(0)}{n!}$. We have to assume that the Taylor series converges toward $f$ in an open interval $(-r,r)$ around zero. Then I have to show that if $a_{2n-1}=0$ for all $n\in \mathbb N$ so is $f(-x)=f(x)$ for all $x\in(-r,r)$. How can I do it? I think I maybe can see on $kx^{2n}$ while all odd joints are zero while $a_{2n-1}=0$? But how can I prove it?