Need to find a Fourier Series... I am to find a Fourier Series for the following function:
$$
y(x)=\sqrt {R^{2}-x^{2}}
$$
about
$$
-R \leq x \leq R
$$
with the recursion
$$
y(x+2R)=y(x)
$$
Do I let$\sqrt {R^{2}-x^{2}}$equal $y$ in this circumstance and proceed to find coefficients $A_0, A_n, B_n$?  

You have an even function of $x$, so you will have a Fourier series of the form
$$y(x) = a_0 + \sum_{k=1}^{\infty} a_k \, \cos{\left ( \frac{\pi k x}{R}\right )}$$
where
$$a_0 = \frac{1}{2R} \int_{-R}^{R} dx \, \sqrt{R^2-x^2} = R \frac{\pi}{4}$$
$$a_k = \frac{1}{R} \int_{-R}^{R} dx \, \sqrt{R^2-x^2} \, \cos{\left ( \frac{\pi k x}{R}\right )} = R \frac{J_1(k \pi)}{k} $$
where $J_1$ is the Bessel function of the first kind of first order.
