fourier analysis - Find flaw in solving for coefficients of $e^x = A_0 + sumlimits_{n=1}^infty A_n cos frac{n pi x}{L}$

$$\begin{align*} e^x &\sim A_0 + \sum\limits_{n=1}^\infty A_n \cos \frac{n \pi x}{L} \\ \end{align*}$$

The Fourier cosine series on the right will be even extended periodized version of $e^x$.

Now, I could solve for these coefficients directly,

$$\begin{align*} A_n &= \frac{2}{L} \int_0^L e^x \cos \frac{n \pi x} \, dx \\ \end{align*}$$

But this exercise asks for a different solution.

We can apply term by term differentiation to yield:

$$\begin{align*} e^x &\sim - \sum\limits_{n=1}^\infty \frac{n \pi}{L} A_n \sin \frac{n \pi x}{L} \\ \end{align*}$$

Setting the two definitions of $e^x$ equal yields the following which is equality for $[0,L]$:

$$\begin{align*} A_0 + \sum\limits_{n=1}^\infty A_n \cos \frac{n \pi x}{L} &\sim - \sum\limits_{n=1}^\infty \frac{n \pi}{L} A_n \sin \frac{n \pi x}{L} \\ \end{align*}$$

From there, how can I derive $A_n$?

EDIT: I found my original flaw, that last equation is only equality for $[0,L]$, not for $[-L,L]$. However, I'm still not sure how to calculate $A_n$ via this technique.